148 Mr Love, Note on Kirchhoff’s theory of the [Nov. 28, 
These are functions of the initial position of Q, ie. of a+a, B+ y, 
and thus the differential coefficient of each of them with respect 
to a must be equal to that with respect to x, and so for 8 and y. 
In forming these differential coefficients it is important to 
observe that uw, v, w are not functions of a, B, for they are the 
relative displacements of a point Q of the prism whose centre 
is P, and the a, 8 are the same for all points of this prism, viz. 
they are the coordinates of P. The differential coefficients of any 
function with reference to a, 8 have reference to the difference of 
the values of the function at corresponding points of contiguous 
prisms, and w, v, w are functions defined with reference to the 
prism whose centre is P. 
Kirchhoff’s equations (Vorlesungen, s. 450) include the diffe- 
rential coefficients which in my notation would be a and these 
are afterwards neglected as small in comparison with = on the 
Ou 
ground that ai will be of the order w, which in the general theory 
of these bodies has been shewn to be negligible in comparison 
a 
with =. 
This is the point in which M. Boussinesq describes Kirchhoff’s 
process as wanting in rigour. He explains the smallness of — 
by saying that at similarly situated points of contiguous prisms 
the difference in the amount of the relative displacements is small 
compared with the difference in the amount of such displacements 
at near points in the same prism, and describes this as an assump- 
tion. We have just given reason for holding that the differential 
du 
Ox 
been defined as functions of position in one elementary prism. 
coefficients such as — do not exist, inasmuch as u, v, w have only 
Omitting these terms the equations obtained are three of the 
type 
ow a) Ow  0& al, 
ol, 
+ y+0)5t4 @+u) St... @). 
x 
and three derived from these by putting y for 2 on the left-hand 
side and @ for a on the right-hand side. 
