VOL. LXXXV.] PHILOSOPHICAL TRANSACTIONS. 577 



14. On multiplying the last 2 series into one another, to obtain a foundation 

 for the demonstration in view, the same powers of x and z, which arise in the 

 multiplication, being placed under one another, the products will stand in a re- 

 gular orderly form, the terms of the same order ranging under each other. 



Now, in order to establish the laws of arrangement on clear and general prin- 

 ciples, it is necessary to observe these particulars. 1st. The exponents of the 

 terms, both in the multiplicand and multiplier, are in arithmetical progression ; 



they have the same denominator r, and r is also the common diiFerence in the 



I 



numerators of each progression. 2d. The multiplicand being multiplied by jr"", 

 the first term in the multiplier, gives the first horizontal lire of products ; and 

 consequently the exponents in this line are obtained from the exponents in the 

 multiplicand by adding 1 to the numerators. The numerators therefore of the 

 exponents of this line are also in arithmetical progression ; and under this the 

 other lines of products are to be arranged, so that terms which have the same 

 exponents may come under one another. 3d. The coefficients being neglected. 



if any term in the multiplicand be denoted by z? jt -■ , the term of the multi- 



plier immediately under will be expressed byz?a,' ' , according to the nature 

 of the two series ; and on multiplying the first term of the multiplicand by this 



term of the multiplier, the product will be z''x •■ , which is equal to that 

 term of the multiplicand immediately over that in the multiplier, after 1 is added 

 to the numerator of the exponent of x. And the other terms in the multipli- 

 cand, successively to the right hand, being multiplied by the same term of the 



multiplier, the terms will be 2'+ 'x •" , z' + ^jt ' , z^ + ^x ' , 

 &c. in arithmetical progression, which are equal to those terms of the multipli- 

 cand immediately over them, after the numerators of the exponents of x are 

 increased by 1 . And hence a general rule is obtained for the arrangement of any 

 horizontal line of products. For when the first term in the multiplicand is 

 multiplied by a term in the multipler, the product is placed immediately under 

 that term of the multiplier ; and the products which arise from multiplying the 

 other terms of the multiplicand, successively towards the right, by the same 

 term of the multiplier, are placed successively towards the right of the first- 

 mentioned product. 



15. The several products therefore, arranged under each other in a perpen- 

 dicular line, arise in the following manner. The first arises from multiplying 

 the term in the multiplicand directly over it into the first term in the multiplier. 



Thus - X „ — X — ;; — z X is the product of - X -^r- X — r— 



VOL. XVII. 4 E 



