178 PHILOSOPHICAL TRANSACTIONS. [aNNO 1697. 



and divide by a, all the products belonging to i^ + ''-'^, except those of the 

 first class; 3. I multiply by d and divide by a all the products belonging to 

 .^m + r-3^ except those of the first and second class ; 4. I multiply by e and di- 

 vide by a all the terms belonging to z"' + '■- *, except those of the first, second 

 and third class, and so on, till I meet twice with the same term. Lastly, I add 

 to all these terms the product of a'" - ' into the letter whose exponent is r -{- \. 

 And here I must take notice that by the exponent of a letter, I mean the 

 number which expresses what place the letter has in the alphabet ; so 3 is the 

 exponent of the letter c, because the letter c is the third in the alphabet. It is 

 evident that by this rule, you may easily find all the products belonging to the 

 several powers of z, if you have but the product belonging to z'", viz. a™. 



To find the unciae which ought to be prefixed to every product, I consider 

 the sum of units contained in the indices of the letters which compose it, the 

 index of a excepted. I write as many terms of the series m X m — \ X m — 



1 X rti — 3, &c. as there are units in the sum of these indices, this series is to 

 be the numerator of a fraction, whose denominator is the product of the several 

 series 1X2X3X4, &c. 1X2X3X4X3, &c. 1 X2X3X4X5 

 X 6, &c. the first of which contains as many terms as there are units in the in- 

 dex o( b, the second as many as there are units in the index of c, the third as 

 many as there are units in the index oi d, the fourth as many as there are units 

 in the index of e, &c. 



Demonstration. — To raise the series az + biz -j- cz^ -f- Jz", &c. to any power 

 whatever, write so many series equal to it as there are units in the index of the 

 power required. Now it is evident that when these series are so multiplied, 

 there are several products in which there is the same power of z; thus, if the 

 series az -}- bzz -f- cz"* -f- di^ &c. be raised to its cube, you have the products 

 /f'z", abcz^, aadz^, in which you find the same power z*. Therefore let us con- 

 sider what is the condition that can make some products to contain the same 

 power of z : the first thing that will appear in relation to it, is that in any pro- 

 duct whatever, the index of z is the sum of the particular indices of z in the 

 multiplying terms ; this follows from the nature of indices ; thus Z;V is the 

 product of bz"^, bz^, bz^, and the sum of the indices in the multiplying terms, is 



2 -f 2 -f- 2 = 6 ; a/jcz" Is the product of az, bzz, cz% and the sum of the in- 

 dices of z in the multiplying terms isl -f-2-|-3=6; aadz'^ is the product of 

 az, az, (/z", and the sum of the indices of z in the multiplying terms is 1 -f- 1 

 -|- 4 = 6 ; the next thing that appears is, that the index of z in the multiplying 

 terms is the same with the exponent of the letter to which z is joined ; from 

 which two considerations it follows that, to have all the products belonging to a 

 certain power of z, you must find all the products where the sum oi the ex- 



