Local Attraction on Geodetic Operations. 255 



Also 



r/=aHl — e(sin'?+ sin' Z';} = | 1 - 1 (2— cos 2l— cos 21') | 



= a^{l— e(l — cosXcos 2m)} ; 

 c2=4a2sin2A| l_e{l + (2+cosX)cos2m} |; 



sm| = ^|l + |{l + (2+cosX)cos2;72}| ; 



= sm-i-^ + i j 1 +(2 + cos X) cos 2m I tau^. 

 2ci 2 \ J 2 



Hence by the first formula, 



5=a^l — ^ «e sin X cos 2m 



=a(2—e) sin-^ — 4-ae{l +(2+ cos X) cos 2m] tan - — ^ae sinXcos2m 

 2(1 2 2 



= (a + 5) sin- 1 ^ + (« - ^ ) I 1 + i ( 1 — cos X) cos 2m j tan |. 



Taking the variation of s with respect to a and b, considering c as constant, 

 and X and m also constant, occurring as they do only in small terms, we 

 shall have the difference in length of two arcs joining the stations and be- 

 longing to different ellipses, only having their axes parallel. Hence 



^ ^ 2a a ^4a'-c' 



+ ^5) I + i(l — cosX)cos2m|tan^. 



Since the terms are small, we may use the first approximate value for c 

 and b ; 



.-. a5=(aa + ^^)--2tan^aa + (aa-g) j 1+ l(l-cosX)cos2ml tan- 

 = {la + lb) - tan ^-^ + (oa - ^5) i tan ^ ( 1 - cos X) cos 2m 



= {la + a5)P + {la - lb) Q cos 2m, 

 where 



P=ix— tanix, and Q=itan^ A(l— cosX) 

 2 2 2 2 



=(P+Qcos 2m)^a + (P— Qcos 2m)lb. 



I will find the values of la and lb which will satisfy this equation and 

 make la^+W a minimum. 



