ON HAMILTON'S NUMBERS. 67 



< >rder 2 p = q z 



t 



V.,\ +UEJM + ...... (3) 



Again, retaining only those terms of (2) whose order is superior to ^, we have 



1> = 3; -fH; -ig + ^+ia*; -a 3 ; -A* 8 .... (4) 

 Order 2 ; f ; 1 ; | ; f . 



From (3) and (4) we obtain by subtraction 



Order $ = $ r 3 - 257- + f 9' 



1 - i ** - 2 ? 'r + V 2 



I + " + 



I +A 3 + |iA/ 



A + 1: B A 



H +f:c 2 



tt + ^ D 311 



M +$E<r'. + ...... (5) 



Changing 7^, q, ,... into 5, J-, s, . . . respectively, equation (4) becomes 



so that, if we assume 7 = r* (1 a), the order of a will be the same as that of r" 8 * 3 , 

 viz., - | + | = - i ; 



Hence, if we substitute ) A (1 a) for q in (5), neglecting in the result quantities of 

 the order ^, we shall find 



|y _ 2qr + |gi - |* - 2 3 '/- + V? 

 = f r - 2/- 3 (l - a) + |- s (l _ | + f 2 + ^ tf ) 

 - i S * - 2^ (1 - $ a) + V s (1 ~ ) 



+ A'V - *** + i'" 8 

 K 2 



