34 Prof. A. Gray on Canonical 



14. It may be noticed here that (1), § 13, gives at once 

 Lagrange's variational equation referred to above. For 

 let 8, 8' denote independent variations. T\ r e have 



8.8'q = 8'.8q, 8.8'$ 1 = 8 , .8S 1 , . . (1) 



and obtain 



S'.SSx = t(8 / p8q-8'b8a-h 1 )8 , .8q-b8'.8a) | 



8. 8 t S l =l t (8p8'q-8b8 , a+p8. 8'q-bS. 8'a)i 



which give by subtraction Lagrange's equation 



X{8'p8q-8p8'q)=2(8'b8a-r-8b8'a). . . (3) 



Thus the quantity on the left is independent of the time t. 



This equation leads at once to the reciprocal relations 

 discussed above in §§6 ... 11. For instance, let all the 

 variations 8 f of the initial coordinates and momenta be zero 

 except 8'a p and all the variations 8 of the final coordinates 

 and momenta be zero except 8qj : we obtain 



Vp^-Sbjl** (4) 



or 



Vp l _ 8b. 



*'«." «fc ( ° } 



15. If in Sj we vary t and r to t + dt and t + 8t we get, 

 by the defining equation for S„ (2) § 4, 



A^ = lj8t-L 8r-i-X(2?8q)-l{b8a), . . (1) 



where A denotes the variation from the value of S x for the 

 times of starting and arrival, t and t, to that for the times 

 t + 8t and t + 8t, and 8q, 8a are variations effected without 

 change of t or t. The insertion of the values of L and Lo 

 gives this the form 



AS 1 =-H8t + B. 8T+X{pAq)-X(bAa), . . (2) 



where Aq — 8q + q8t, Aa = 8a + a8t. 



Taking successive variations A, A' and again A', A, and 

 subtracting we get an extension of Lagrange's theorem in 

 the forms 



8'L8t-8L8't+t(8 , p8q-8p8 , q) 



= 8 / L 8T-8L 8'T + Z(8'b8a-8L8'a), . . (3) 

 and 



t(A'pAq-ApA'q) + A'mt-AE.8't 



= t(A'bAa-AbA'a)+A'H 8T-AH 8'T. (4) 



