150 Archdeacon Pratt on the Anglo -Gallic, Russian, 



3. I have now to find, if possible, values of the local attrac- 

 tions which wilt inake these three ellipses, representing the 

 Anglo-Gallic, Russian, and Indian arcs, the same— that is, 

 a x =a^aQi and b l = b 2 =b 3 . These lead to the following equa- 

 tions : — 



1944(0 + 2122(0- 686 = 0, 



7254(0 -17414(/ 2 ) -5050=0, 



1944(0-87353(0 - 562=0, 



7254(0 +35007(0 -4768=0, 

 or 



(3-2886963) ft) + (3-3267454) (O- (2-8363241) = 0, 



(3-8605776) (O - (4-2408985) (O - (3" 7032914) =0, 



(3-2886963)(0 - (4-9412778) (0~ (2*7497363) = 0, 



(3-8605776)(O + (4-5441549) (0- (3-6783362) = 0. 



By the method of least squares we are able to obtain the most 

 likely values of (0, (0> fe) which satisfy these four equations. 

 For this end, multiply each equation by the coefficient of (0, 

 and add all together; do the same for (0 and (0 ; and we have 

 three equations, the solution of which gives the required quan- 

 tities. 



For the first equation, 



(6-5773926)(0 + (6-6154417) (O -(6- 1250204)-} 



+ (7-7211552)(0 -(8-1014761) (O - (7-5638690) I _ n 

 -f(6-5773926)(0-(8-2299741)(0- (6-0384326) [ ; 

 + (7-7211552) (O + (8-4047325) {t 3 )~ (7-5389138) J 



or, rejecting 3 from the indices throughout, 



3779-| (0+ 41251 (^)-169814-1(0- 1334 



52621 I — 126321 J + 253941 J -36633 



. 3779 | - 1093 



52621J -34587 



112800 (O -122196 (0+ 84127 (0-73647 = 0, 

 or 



(5-0523091) (0 - (5-0870570(0 + (4-9249354) (O 



- (4-8671551) =0 (a) 



For the second equation, 



(6-6154417) (0 + (6-6534908) (0 - (6*1 630695) \ ' 

 + (8-1014761) (O - (8-4817970) (O -(7-9441899) J ' "" 



