ON HAMILTON'S NUMBERS. 69 



In the second line of (6) we may reject the whole of a', since its order is ^, and 

 write 



After substituting then- values for the terms in (6) which contain a, and at the 

 same time dividing throughout by f , we shall obtain 



Order 1 = |r-V - s* + r 8 



| + 



f + r 



A 



We now write 



r = l-8 and = *- 



where, observing that the values of /8 and /8' can be immediately deduced from those 

 of a and a' by changing r, s, t, . . . into , t, u, . . . , it is evident that and ft are 

 both of the order ; for a and of are both of the order . Thus (neglecting 

 quantities whose order is equal to, or less than, ) we have 



Order |; f ; A ; tf 



Order 



