9 
the ‘Differential Calculus’ of Du Bourguet/’ by Sir 
James Cockle, F.B.S., Corresponding Member of the So- 
ciety. 
In “Notes and Queries” for September 6, 1879 (5th S., 
vol. xii., pp. 182-3) I have given a bibliography of Du 
Bourguet’s work* on the calculus (Paris, 8vo; vol. i., 1810; 
vol. ii., 1811). From that work (viz,, from vol. ii., pp. 75-0) 
I translate the following article, premising that Du Bour- 
guet’s equation (at p. 75) is the well-known criterion of 
integrability : — 
“361. 2°. Every equation between three variables which 
does not satisfy this (330), is not integrable, and consequently 
appears at first sight destitute of significance. Nevertheless M. 
Monge has demonstrated, in the Memoirs of the Academy of 
Sciences of Paris (year 1874), that such an unintegrable differen- 
tial equation, between three variables, represents an infinite 
number of curves of double curvature possessing a common 
property. Besides, we shall observe that, in these non-integrable 
equations, this (330) gives a relation between the three variables, 
which, as it stands, or augmented, or diminished by a constant 
quantity, often satisfies the proposed ; such is, for example, the 
equation 
(y - z)dz + {z - y)dx + (^i? + a)dy = 0 . . . . (a), 
for which the equation of condition (330) is not satisfied, since it 
is in this case 
z =■■ X y -{■ a 
yet if we substitute this value of z in the proposed (a), we have 
the identical equation 0 = 0. 
“ Again, let there subsist 
{1 ^ {z - y - x)[\ a^ {z - y - x) - {z-y - x)]]dx 
+ [1 + x^ i^-y- so)\dy -dz = 0 (6), 
for which the equation (330) becomes 
/3a\i2 
z = x + W- 
* I first saw a copy of this work in July last, when my brother. Dr. John 
Cockle, presented me with the two uncut volumes, bound in a printer’s 
covering or binding of blue paper, and each with a white ticket on the back 
whereon the words “ Du Bourguet. Calc. Dilf. et Integral ” are printed, 
with the respective additions “Tome I.” and “Tome II.” 
