244 Prof. F. Y. Edgeworth on the Application of 



Transform the system by a linear (preferably orthogonal) 

 substitution so that T the kinetic energy assumes the form 



B 1 n 1 ' 2 + B 2 n 2 2 + ...+w+W + ...; • • CO 



where the B's and b's are functions of the co-ordinates, the 

 IPs and 7r's are " momentoids." The general expression 

 for IIx', what Ili becomes in consequence of a collision, in 

 terms of the IPs, viz. H 1 — A X B, then reduces to 



, ( AiBJi + A 2 B 2 n 2 + . . . -\ x b 1 ir 1 -X 2 6 2 7r 2 - . . . 



111 " x x A^Bi + A 2 2 B 2 + . . . + V^i + V&s + • • • 



where A 1? A 2 ...X 1? X 2 ... correspond to the L 1; L 2 .../ 1? / 2 ... 

 etc. as defined in the general case (I. c. p. 262) ; and 

 B 1; B 2 . ..b 1 ... correspond to B n , B 22 ...& n ... ; except that the 

 momentoids are not (in general) true (Lagrangian) com- 

 ponents of momentum (a circumstance which does not affect 

 the present argument concerned only with collisions and 

 the attendant impulses). Forming the mean square for 

 components of momentum (the mean obtained by supposing 

 the velocities to acquire all possible values while the co- 

 ordinates and points of impact remain the same), and equating 

 [IT/ 2 ] to [II 1 2 ], and remembering that the mean products- 

 [YI r II J vanish, we obtain 



-4A 1 2 A 2 2 (B 1 B 2 [n 1 2 ]-B 2 2 [n 2 2 ]1 



-4A 1 2 A 3 2 (B 1 B 3 [II 1 2 ]-B3 2 [n 3 2 ] 



-4A 1 2 Xi 2 (BA[n 1 2 ]-V[ 7 r 1 2 ] 

 -4A 2 2 X 2 2 (B 1 6 2 [n ) 2 ] -6 2 ' 2 [tt 2 2 ] 



h=0. 



J 



Now it is known (by 1. 1) that the sought frequency-curve- 

 is of the form 



Const, exp - (C l n 1 2 +C £ n 2 2 +... + c^ + c^ + ...)- 

 Put C ] = K 1 B 1 , C 2 = K 2 B 2 , ... Cl = ^A; 



and the last written equation becomes 



- 2A 1 2 A/(B 1 /K 1 -B 2 /K 2 )-2A 1 2 A 3 2 (B 1 /K\-B 8 /K 3 ) . . . 

 -2A I % 2 (B 1 /K 1 -V«-...=0. 



Since this equality holds good for all the values which 

 the A's and X's can assume as the points of contact vary 

 (the co-ordinates remaining the same), it follows that 



K T = K 2 = K 3 . . . = l\ = k 2 . 



