204 Transactions. — Miscellaneous. 



It is evidently desirable that the corrections should be as 

 small as possible so that no undue alterations are made to the 

 angles : this condition is satisfied when the sum of the squares 

 of the corrections is a minimum. 



The application of this condition is shown on the schedule, 

 and is briefly as follows : — 



In column No. 5 the natural sines of the angles in column 

 No. 4 are given. 



If the sines in No. 5 were correct we should have — 



Sin Ai sin A 2 sin A 3 sin A 4 -. 



Sin Bj sin B 2 sin B 3 sin B 4 ~~ 



This equation shows that the length of P P 2 calculated from 

 P P 4 by the first pair of triangles should be the same as the 

 length calculated by the second pair of triangles. 

 This is not usually the case, so put 



Sin A x sin A 2 sin A 3 sin A 4 



= 1 + 



€. 



Sin Bi sin B 2 sin B 3 sin B 4 

 where the sines are taken from column No 5 and c is n 

 radians. 



To convert e into seconds multiply the value in radians by 

 206265 (= number of seconds in 1 radian). 



The calculation is shown on the schedule, giving, in this 

 particular example, e = + 28"01. 



(Note. — Attention must be paid to the sign of e.) 



The other necessary condition is that the sum of the 

 angles Cj and C 2 should equal the sum of the angles of C 3 and 

 C 4 , or— 



C x + 2 = C 3 + d. 

 This is not usually the case, so put 



Ci + C 2 = C 3 + C 4 + € , 



where the angles are taken from column No. 4 and e is in 



seconds. This gives e = — 5" in this example (see schedule). 



(Note. — Attention must be paid to the sign of e .) 



In column No. 6 the natural cotangents of the angles are 

 inserted. 



In column No. 7 twice the cotangents are entered. 

 Let a x = cot Aj 

 „ ft = cot B x 

 and similarly for the other angles. 

 Let a x = 2d! + ft 

 „ 6, = - a x - 2ft 

 » C\ = — <*i + ft 

 and similarly for a 3 , b 3 , c 3 : a t , b it c 4 . 



In column No. 8 the values of a lt b lt c x , &c, are given, and 

 a check is obtained by noting that a t + b x + c x = o. 



Square all the values in column No. 8 and add them. This 



