181 
“Oa a Proposition of Du Bourguet,” by Sir James 
Cockle, F.E.S., Corresponding Member of the Society. 
The remarkable passage which (Sup., pp. 9 — 10) I have 
translated from Du Bourguet conflicts with reiterated 
statements of Lagrange, who (Legons, 1806, pp. 340 — 1), 
speaking of a certain equation, which is in substance the 
same as Du Bourguet’s (380), and which expresses the crite- 
rion of integrability, says 
“ And this equation ought to hold in itself, that is to say, 
“ to be identical, in order that the variable 0 may be capable 
“ of being a function of x and y, and that, consequently, the 
“ one proposed may have a primitive in x, y, and 0 .” 
Moreover, Lagrange says (ih. p. 841) 
But when the equation of condition does not hold in 
“itself, the proposed equation cannot subsist unless we 
“ assume some relation between x, y, and 0 , in such manner 
“ that two of these variables become functions of the third.” 
Again, speaking of a certain case in which the condition 
is not satisfied, Lagrange (ih. p. 342) says 
“ Thus it is impossible that 0 can be a function of x and y 
“ considered as independent of each other.” 
Still Du Bourguet’s conclusion from his example (a) may 
be easily verified. Under these circumstances, when we 
find Du Bourguet writing the following paragraph (vol. i., 
p. xxiij.), 
“ At the end of each volume I have placed a Table recapi- 
“ tulatory, article by article, of the subjects contained in the 
“volume; and in order to save accomplished mathemati- 
“ cians, who have scarcely the time to read the whole of an 
“ elementary Treatise, the trouble of seeking for the articles 
“ which may be of interest to them, I have marked in the 
“ Tables recapitulatory of the two Parts, the articles the 
“ most worthy of the attention of these mathematicians, by 
“ one or two asterisks, according to the importance of the 
“ subject.” 
