gives a genuine representation of the country which it is intended to deal with, so that after a few trials, a centre line can be laid down, which will require no further alteration. Longitudinal and cross-sections can be made; and the results of the cubications ought to vary by only a very small percentage from the definite quantities eventually found by means of the working plans.

In Plate 5, Fig. 10, a piece of country is represented mapped out in this way; and Plate 5, Fig. 11, shows the difference between the preliminary and the definite working section of the centre line adopted. This is a case of actual practice on a railway now being constructed. The sample field-book contains some of the operations carried out for forming this plan.

CONCLUSION.

Many engineers imagine that tacheometry involves a plicated measuring instrument, a number of peculiar accessories, and tedious calculations in the office; but it is hoped that the foregoing explanations will have shown the contrary; for, with an instrument, which is nothing but a transit theodolite, a staff, and simple tables, preliminary plans are produced more rapidly, and giving infinitely more detail, than by the old methods. No doubt a great deal of the suspicion with which this system has hitherto been looked upon was due to the peculiar graduation of 400 degrees to a circle, by which all existing trigonometrical tables are rendered useless; but a careful inquiry at once shows that this is not one of its integral parts. Whenever this fact is generally admitted, a radical change is bound to take place in the make of theodolites. Engineers ought never, in the future, to purchase these as ordinarily constructed, but to insist on the makers supplying them with distance-measuring telescopes, such as are fitted to the present tacheometers. Then tacheometry, which is now looked upon as a special branch of surveying, will pass into one of the ordinary uses of the theodolite.

The Paper is illustrated by four tracings and a photograph, from which Plate 5 has been made.

[APPENDIXES.

0 1.0000 2.0000 3·0000 4.0000 50000 60000 7·0000 8.0000 9.0000 60 10 1.0000 2·0000 3·0000 4.0000 5·0000 60000 70000 8·0000 9.0000 50 20 1.00002 0000 3·0000 4.0000 5·0000 60000 70000 8·0000 9.0000 40 30 0.9999 1.9998 2.9997 3.9996 4·9995 5 9994 6·9993 7·9992 8 · 9991 30

0.0017 0.0035 0·0052 0·007 0.009 0.010 0.012 0.014 0.015 7 0.00200·0041 0·0061 0.008 0.010 0.012 0.014 0.016 0.018 8 0.0023 0·0047 0·0070 0.010 0.012 0.014 0.016 0.019 0.021 9 0.0026 0·0052 0·0079 0.011 0.013 0.016 0.018 0.021 0.024 10 0.0029,0·0058 0·0087 0.012 0.014 0.017 0.020 0.023 0.026

11 0.0032 0·0064 0·0096 0.013 0.016 0.019 0.022 0.025 0.029 12 0.0035 0·0070 0·0105 0.014 0.017 0.021 0.024 0.028 0.031 13 0.00380 0076 0·01140.015 0.019 0.023 0.027 0.030 0.034 14 0.0041 0·0081 0·0122 0·016 0.020 0.025 0.029 0.033 0.037 15 0.0044 0.0087 0·0131 0·017 0.022 0.026 0.031 0.035 0.039

16 0.00470 0093 0·01400 019 0.023 0.028 0.033 0.037 0.042 17 0.0049 0·0099 0.0148 0.020 0.025 0.029 0.035 0.039 0.044 18 0.0052 0.0105 0·0157 0.021 0.026 0.031 0.037 0.042 0.047 19 0.00550 0111 0·0166 0.022 0.027 0.033 0.039 0.044 0.050 20 0.0058 0.01160 0174 0.023 0.029 0.035 0.041 0.046 0.052

21 0.00610 0122 0.0183 0.024 0.030 0.037 0.043 0.049 0.055 22 0.00640 0128 0·0192 0.026 0.032 0.038 0.045 0.051 0.058 23 0.0067 0·0134 0·0201 0.027 0.033 0.040 0.047 0.054 0.060 24 0.00700 0140 0·0209 0.028 0.035 0.042 0.049 0.056 0.063 25 0.0073 0·0146 0·0218 0.029 0.036 0.044 0.051 0.058 0.065

26 0.00760·0151 0·02270.030 0.038 0.046 0.053 0.060 0.068 27 0.0078 0·0157 0·0235 0.032 0.039 0.047 0.055 0.063 0.071 28 0.0081 0·0163 0·0244 0.033 0.041 0.049 0.057 0.065 0.073 29 0.0084 0·0169 0·0253 0.034 0.042 0.051 0.059 0.068 0.076 30 0.0087 0·0174 0·0262 0.035 0.044 0.053 0.061 0.070 0.079

54 0.99999 53 0.99999 52 0.99999 51 0.99999 50 0.99999

49 0.99999

48 0.99999 47 0.99999 46 0.99999 45 0.99998

44 0.99998 43 0.99998 42 0.99997 41 0.99997 40 0.99996

39 0.99996 38 0.99996 37 0.99996 36 0.99995 35 0.99995

34 0.99994 33 0.99994 32 0.99993 31 0.99993 30 0.99992

30 0.9999 1.9998 2.99973.9996 4 9995 5 9994 6.99937.9992 8.9991 30 40 0.9999 1.9998 2.9997 3 9996 4 99955 9994 6.9993 7·9992 8.9991 20 50 0.99981.9996 2.9994 3.99924-9990 5 9988 6·9986 7·99848-9982 10 60 0.99971.99942-99913.99884-9985 5 9982 6·9979 7·99768.9973

0

30 0.0087 0.0174 0.0262 0.035 0.044 0.053 0.061 0.070 0.079 30 0.99992 31 0.0090 0·0180 0.0270 0.036 0.045 0.054 0.063 0.072 0.081 29 0.99992 32 0.0093 0.0186 0·0279 0·037 0.046 0.056 0.065 0.074 0.084 28 0.99991 33 0.00960 0192 0.0288 0.038 0.048 0.058 0.067 0.077 0.086 27 0.99991 34 0.0099 0.0198 0.0297 0·040 0.049 0.059 0.069 0.079 0.089 26 0.99990 35 0.01020 0204 0·0306 0.041 0.051 0.061 0.071 0.082 0.092 25 0.99990

36 0.0105 0.02100-0315 0.042 0.052 0.063 0.073 0.084 0.094 37 0.01080 02160·0324 0.043 0.054 0.065 0.076 0.086 0.097 38 0.0111 0·0222 0·0332 0.044 0.055 0.067 0.078 0.089 0.100 39 0.0113 0.0227 0·0340 0·045 0.056 0.068 0.080 0.090 0.102 40 0.01160 0232 0.0348 0.046 0.058 0.070 0.082 0.093 0.104

41 0.01190·0238 0·0357 0.048 0.059 0.071 0.084 0.095 0.107 42 0.0122 0.0244 0·0366 0.049 0.061 0.073 0.086 0.098 0.110 43 0.0125 0.0250 0·0375 0.050 0.063 0.075 0.088 0.100 0.112 44 0.0128 0·0256 0·0384 0.051 0.064 0.077 0.090 0.102 0.115 45 0.0131 0·0262 0·0393 0.052 0.065 0.079 0.092 0.105 0.118

46 0.0134 0·0268 0·0402 0.054 0.067 0.080 0.094 0.107 0.121
47 0.0137 0.0274 0.0411 0.055 0.068 0.082 0.096 0.110 0.123
48 0.0140 0·0280 0·0420 0.056 0.070 0.084 0.098 0.112 0.126
49 0.0143 0·0286 0·0429 0.057
50 0.01450·0291 0·0437 0.058

0.071 0.086 0.100 0.114 0.129
0.072 0.087 0.102 0.116 0 131

51 0.0148 0·0296 0·0445 0.059 0.074 0.089 0.104 0.118 0.133 52 0.0151 0·0302 0·0453 0.060 0.075 0.091 0.106 0.121 0.136 53 0.0154 0·0308 0·0462 0 062 0.077 0.092 0.108 0.123 0.139 54 0.01570 0314 0.0471 0.063 0.078 0.094 0.110 0.126 0.141 55 0.01600 0320 0·0480 0.064 0.080 0.096 0.112 0.128 0.144

56 0.0163 0·0326 0·0489 0.065 0.081 0.098 0.114 0.130 0.147 57 0.0166 0.0332 0.0498 0.066 0.083 0.100 0.116 0.133 0.149 58 0.0169 0.0338 0·0506 0.068 0.084 0.101 0.118 0.135 0.152 59 0.01720-0343 0·0514 0.069 0.086 0.103 0.120 0.137 0.154 60 0.01740 0348 0·0522 0·070 0.087 0.104 0.122 0.139 0.157

[THE INST. C.E. VOL. XCIX.]

89 degrees.

2500 6.0000 6·7500 60
2325 5·9800 6.7275 50
2143 5·95926-7041 40

0 0 7500 1.5000 2·2500 30000 3·7500 4.5000 5
10 0.7475 1·4950 2·2425 2·9900 3·73754 4850 5
20 0.7449 1·4898 2·2347 2·9796 3·7245 4 4694 5
30 0.74241·4848 2·2272 2·9696 3·7120 4·4544 5 1968 5·9392,6-6816 30

0 0.43300 8660 1 2990 1.732 2.165 2.598 3.031 3.464 3.897 1 0.4331 0·8663 1·2994 1.732 2.165 2.599 3.032 3.465 3.898 2 0.4333 0.8666 1·2999 1.733 2.166 2.600 3.033 3.466 3.900 3 0.4334 0.8669 1·3003 1.734 2.167 2.601 3.034 3.467 3.901 4 0.43360 8672 1.3008 1.734 2·168 2.602 3.035 3.469 3.902 5 0.4337 0.8675 1.30121.735 2.168 2.602 3.036 3.470 3.903

6 0.4339 0·8678 1·3017 1.736 2.169 2.603 3.037 3.471 3.905 7 0.43400 8680 1.3021 1.736 2.170 2.604 3.038 3.472 3.906 8 0.4341 0 8683 1 3025 1.737 2.170 2.605 3.039 3:473 3.907 9 0 43430 8686 1 3029 1.737 2.171 2.606 3.040 3.474 3.909 10 0 43440 8689 1·3033 1.738 2.172 2.607 3.041 3.475 3.910

14 0 4351 0·8700 1·3050 1.740

15 0.4352 0 8703 1 3055 1.741

11 0.4346 0·8692 1·3037 1.738 2.173 2.608 3.042 3.477 3.911
12 0.4347 0·8694 1-3041 1.739 2.173 2.608 3.043 3.478 3.912 48
13 0.43490 8697 1.3045 1.740 2.174 2.609 3.044 3.479 3.914 47
2.175 2.610 3.045 3.481 3.916
2.176 2.611 3.046 3.482 3.917 45

49

0.74722

0.74697

0.74672

46

0.74656

0.74621

16 0.4353 0 8706 1·3059 1.741

44

0.74596

0.74570

0.74545 0.74520 0.74494

2.176 2.612 3.047 3.483 3.918 17 0.4355 0.8709 1·3064 1.742 2.177 2.613 3.048 3.484 3.919 43 18 0.43560 8712 1 3068 1.742 2.178 2.614 3.049 3.485 3.920 42 19 0.4357 0·8715 1·30721.743 2.178 20 0 43590-87181.30771-744 2 179

2.614 3.050 3 486 3.921 41
2.615 3.051 3.487 3.923 40

21 0.43600 8721 1.3081 1.744 2.180 2.616 3.052 3.488 3.924 39 22 0.4362 0 8724 1·3086 1.745 2.181 2.617 3:053 3.489 3.926 38 23 0.4363 0·8727 1 3090 1.745 2.181 2.618 3.054 3.490 3.927 37 24 0.4365 0·8730 1·3094 1.746 2.182 2.619 3.055 3.492 3.928 36 25 0.4366 0·8733 1·3099 1.746 2.183 2.620 3.056 3.493 3.929 35

26 0.4367 0.8735 1.3103 1.747 2.184 2.620 3.057 3:494 3.930 34 27 0.4369 0·8738 1·3107 1.747 (2·184 2.621 3.058 3.495 3.932 33 0.74317

28 0.43700 8740 1.3111 1.748 2.185 2.622 3.059 3.496 3.933 32 0.74291

29 0:4371 0·8743 1.31151.748 2.185 2.623 3:060 3.497 3.934 31 0.74266 30 0.43730 8746 1.3119 1.749 2.186 2.624 3.061 3.498 3.936 30 0.74240

Sine2.