576 
Proceedings of the Royal Society 
Without saying anything as to the verification of the develop- 
ments thus obtained, Laplace concludes as follows : — 
“ On decomposeroit de la meme manikre liquation R en 
termes composes de facteurs de 4, de 5, &c., dimensions.” 
To show how this could he effected would have been a tedious 
matter, if the method of exposition used in the previous cases had 
been followed, viz., multiplying instances with wearisome iteration 
of language until the laws for the combination of the letters could 
with tolerable certainty be guessed. On the other hand, had Laplace 
condensed his directions in the way we have indicated, the rule for 
the case in which as many as possible of the factors are of the 4th 
degree could have been stated as simply as that for either of the two 
cases he has dealt with. The only changes necessary, in fact, are 
in parts (1) and (4), and merely amount to writing the letters in 
consecutive sets of four instead of two or three. 
Further, when the rule is condensed in this way, the problem of 
finding the number of terms in any one of the new developments — 
a problem which Laplace solves in one case by considering how 
many terms of the final development each such term gives rise to — 
is transformed into finding the number of possible arrangements 
referred to in part (1) of the rule. Where the highest degree of 
the factors of each term is 2 and the resultant which we wish to 
develop is of the 2nd degree (which is the case Laplace takes), the 
number of such arrangements is evidently (1.2.3 ....%)/ (1.2) s , 
s being the highest integer in n/ 2 ; if the highest degree of the 
factors is 3, the number of arrangements is 
1.2.3 n 
(1.2.3) 8 (1.2)* ’ 
where s is the highest integer in n/3 and t the highest integer in 
(n - 3s) I "2 ; and so on. 
The facts in reduction of the claim which Laplace has to the 
expansion-theorem now bearing his name are thus seen to be (1) 
that the case in which as many as possible of the factors of the terms 
of the expansion are of the 2nd degree had already been given by 
Vandermonde ; (2) that Laplace did not give a statement of his rule 
in a form suitable for application to all possible cases, and, indeed, 
was not sufficiently explicit in the statement of it for the first two 
