280 . PHILOSOPHICAL TBANSACTIONS. [aNNO 1758. 



px, qxy rx, sx, &c. successively for x, in the given value of s, and then dividing 

 the sum of all the quantities thence arising by the given number n. 



The same method of solution holds equally, when, in taking every wth term 

 of the series, the operation begins at some term after the first. For all the 

 terms preceding that may be transposed, and the whole equation divided by the 

 power of X in the first of the remaining terms ; and then the sum of every wth 

 term, beginning at the first, will be found by the preceding directions; which 

 sum, multiplied by the power of x that before divided, will evidently give the 

 true value required to be determined. Thus, for example, let it be required to 

 find the sum of every 3d term of the given series a -^ hx -\- cx"^ -{■ dx^ -f- ex*, 

 &c. (= s), beginning with cx^. Then, by transposing the first two terms, and 

 dividing the whole by x"^, we shall have c -\- clx -{- ex^ -{■ fo^ &c. = ^ ~ " ~ ^"^ 

 (= s'). From which having found the sum of every 3d term of the series c -f 

 dx -^ ex^ -\- fx^ &c. beginning at the first c, that sum, multiplied by a^^, will 

 manifestly give the true value sought in the present case. 



And here it may be worth while to observe, that all the terms preceding that 

 at which the operation, in any case, begins, may (provided they exceed not in 

 number the given interval n) be entirely disregarded, as having no effect at all 

 in the result. For if in that part (— — '^^— ) of the value of s', above exhibited, 

 in which the first terms, a and bx, enter, there be substituted px, qx, rx, suc- 

 cessively, for 07, according to the prescript, the sum of the quantities thence 

 arising will be - ~ - ^ - ;:^, - ^ - ^^ - j^, which, because p' = i, 

 ^^ = 1, &c. or jb^ =^, q^ = -'-, &c. may be expressed thus: 

 -£ X (P + 9 + r) -^X{p' + q' + r'). But, that/, + 9 + r = O, and 

 />* + 9^ + ^'^ = ^3 has been already shown ; whence the truth of the general 

 observation is manifest. Hence it also appears, that the method of solution 

 above delivered, is not only general, but includes this singular beauty and advan- 

 tage, that in all series whatever, the terms of which are to be taken according 

 to the same assigned order, the quantities (jb, 9, r, &c.), by which the solution 

 is performed, will remain invariably the same. The greater part of these quan- 

 tities are indeed imaginary ones; and so likewise will the quantities be that result 

 from them, when substitution is made in the given expression for the value of s. 

 But by adding together, as is usual in like cases, every two corresponding values, 

 so resulting, all marks of impossibility will disappear. 



If, in the series to be summed, the alternate terms, viz. the 2d, 4th, 6th, 

 &c. should be required to be taken under signs contrary to what they have in 

 the original series given ; the reasoning and result will be nowise different ; only 

 instead of making p^ -\- q^ + 7^, or p -\- r/" + r", &c. = + 3. or -]- n, the 



