318 
MR. R. S. HEATH ON THE DYNAMICS OF 
55. For brevity denote 
£(»+£ y+y)bJ ©, 
3(x—£, y—y) by 0', 
$(x, y) by 3, 
y) b y 0, 
for all the different suffixes. Also let 
C 0 =['^o( a O 2/)].r=o,y=05 & c - 
for all the even functions. 
Mr. Forsyth finds formulae for the products © ©' for different values of the suffixes. 
These are given on p. 834 et seq. of his Memoir. In the 13th set occur the first four 
of the following six formulae. The remaining two formulae are not in his list, but are 
proved in exactly the same way. 
C 4 C 8 ©0 © 12= ^0 ^12"h ^1 ^13'^5'^9 ^6 ^kA^14 ^7 ^11^3 ^15 
C 0 C 2 ©14© 12= '^14'^12 - t - ^14^12^0-^2 ^5^7 ^9^11 ^9 @1V% 
C 2 C 9 ©7 © 12= ^2^9 ^7 '^12~b^7 ^12^2'^9 - b^5^14'^0'^ll”b ^0 ^11^5 -^14 
C 9 f 12©9 © 12=^9^12^9 '^12 - b^0 ^5 ^0^5 ^2^7 ^2^7 ^11 ^14^11^14 
C 2 C 12©2 © 12=^2^12^2 ’^'12'h^O ^o^lA^ll ~F ^9 ^7 ^9 ^7 
C 0 C 9 ©5 © 12= ^0^9 ^5 ^12 H - ^7 ^14^Al 1 "F ^5^12'^0’^9 "F ^2 ^11^7 ^14 
If in each of these formulae we change the sign of F- and then subtract each new 
formulae from that from which it is derived, we deduce the following six equations :— 
C 4 C 8 (©0 © 12 © 0 © 12 ) = '^ A @13^5^9 ^G^IO^Ah) 
C U C 2 (©14© 12 © 14©12) =’^ (^9^11^5^7 ) 
C 2 C 9 (©7 © 12 © 7 © 12)=’^(^7 ^ 12 ^ 2'^9 ~F ^ O ^ lA ^ Tt ) 
C 9 Ci 2 (© 9 © 12 ©9 © 12 ) = -A 0 5 3q 3- OcjOrj 3^ ) 
C 2 C 12(®2 © 12 © 2 © 12 ) = ’“ A ^14^0'^14 - t _ ^7^9 ^7^9 ) 
C 0 C 9 (®5 © 12 © 5 © 12 )= “A ^12^0^9 “F ^2^11^Au) 
56. The odd functions 3 5 ,.. . will vanish when x=Q, y— 0. Let c 5 be the coefficient 
of x in the expansion of 3-, so that 
and so on for all the odd functions. 
Differentiate all the equations of the last set with respect to £ and then put £=0. 
We notice that 
