VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 179 



ponents of the letters which compose them, shall always be the same with the 

 index of that power. Now this is the method I use to find easily all the pro- 

 ducts belonging to the same power of z: let m + r be the index of that power; 

 I consider that the sum of the exponents of the letters which compose these 

 products, must exceed by 1 those which belong to 2: *■ ~ ' ; now because the 

 excess of the exponent of the letter b, above the exponent of the letter a, is 1, 

 it follows that if each of the products belonging to z'« + ''- ' be multiplied by b, 

 and divided by a, you will have products, the sum of whose exponents will be 

 m + r; likewise the sum of the exponents of the letters which compose the 

 products belonging to z"" + "■, exceeds by 1 the sum of the exponents of the 

 letters which compose the products belonging to z"^+^—^; now because the 

 exponent of the letter a is less by 2, than the exponent of the letter c, it fol- 

 lows, that if each product belonging to z"* + *■ ~ ^ be multiplied by c and divided 

 by a, you will have other products, the sum of whose exponents is still m -\- r; 

 now if all the products belonging to z'" + »•- 2 were multiplied by c and divided 

 by a, you would have some products that would be the same as some of those 

 found before, therefore you must except out of them those that I have called 

 products of the first class. What has been said shows why all the products be- 

 longing to z™ + '■~^, except those of the first and second class, must be multi- 

 plied by d and divided by a : lastly, you see the reason why to all these products 

 is added the product of a"" + ' by the letter whose exponent is r -|- 1 ; which is 

 because the sum of the exponents is still w -j- r. 



As for what relates to the unciae; observe that when you multiply oz -|- bzz 

 4- cz" -1- dz* &c. by az + bzz + cz^ + dz* &c. each letter a, b, c, d, &c. of the 

 second series, is multiplied by each of the letters a, b, c, d, &c. of the first 

 series ; thus the letter a of the second series is multiplied by the letter b of the 

 first, and the letter b of the second series is multiplied by the letter a of the 

 first; therefore you have the 2 planes, ab, ab or 2nb; for the same reason you 

 have lac, lad, &c. Therefore you must prefix to each plane of those that com- 

 pose the square of the infinite series az + ^^z + ^z^ &c. the number which 

 expresses how many ways the letters of each plane may be changed; likewise if 

 you multiply the product of the two preceding series by az -f- bzz -\- cz' &c. 

 each letter a, b, c, d, of the third series is multiplied by each of the planes 

 formed by the product of the first and second series ; thus the letter a is multi- 

 plied by the planes be and cb ; the letter b is multiplied by wcand ca ; the letter 

 c is multiplied by ab and ba ; therefore you have the 6 solids, abc, acb, bac, 

 bca, cab, cba, or 6 abc ; therefore you must prefix to each solid whereof the cube 

 of the infinite series is composed, the number which expresses how many ways 

 the letters of each solid may be changed. And generally, jou mubt prefix to 



A A 2 



