20 
]ME. W. SPOTTISWOODE OX AX EXTEXDED FOE3I OF 
Hence 
Vl^^= 2 {(??' — ^’+ l)s + (4 + 1)2“'* } 
\j\u = 2 { (w — ^ + i)s~* + (*’ + 1)2^ } «i^~y , 
V“m=2{(w— *+ 1)2 *+(*+l)s‘^*|'”(*,*i’” y, 
where the operative factor must be understood to affect the suffixes of the «s and the 
*s in the coefficients, and not the indices of the variables. In the use of these formulae, 
however, it is to be observed that the order of the multiplications must be retained. 
Thus : 
• —12 
|(?* — * + 1)2 +(* + 1)2*1 Ui 
= {(?* — *+1)2 *(** — *’+1)2 * 
_d _£ 
+ (** * + 1)2 *(* + 1)2* 
A- _A 
+ (* + lj2*(** — * + 1)2 * 
d_ d 
+ (* + l)2*(* + l)2* 
= {(**— «+l)(**— *+2)2~^* + (**—*+l)*+(**—«)(*+l) +(*+!)(*■+ 2)2^* I 
= (** — *+!)(** — *+2)«;_2 + {(*2- — *+l)*+(** — *)(* + l)|a,+(« + l)(*+2)(*,_2. 
Similarly, since 
{a-\-hf = a a a 
-\-h a a 
ab a 
a ah 
Therefore 
{(** — *+l)2“*+(* + l)2* 
=[](**— *+!)(**— *+2)(**—*+ 3)2"^* 
ah h 
h ah 
h h a 
Afhhh. 
+ {(**— *+l)*(*+l)+(*+l)(?^—*)(*+l)-|-(*+l)(^4-2)(^*—? — l)|2* 
+ (*+l)(«+2)(*+3)2'*]«i 
