NOTES AND REFERENCES. 
583 
I recall the fundamental idea of Lejeune-Dirichlet’s investigation ; starting with an 
integral 
over a given volume he replaces 
where p 
is a discontinuous function, = 1 for points inside, and = 0 for points outside, the given 
volume ; such a function is expressible as a definite integral (depending on the form 
of the bounding surface) in regard to a new variable 6 : the limits for x, y, z, may 
now be taken to be oo, — oo for each of the variables x, y, z, and it is in many 
cases possible to effect these integrations and thus to express the original multiple 
integral as a single definite integral in regard to 9. 
I have not ascertained how far the wholly different method in Lejeune- 
Dirichlet’s Memoir “ Sur un moyen général de vérifier l’expression du potentiel relatif 
à une masse quelconque homogène ou hétérogène,” Crelle, t. xxxn. (1846), pp. 80—84, 
admits of extension in regard to the number of variables, or the exponent of the 
radical. 
4. As noticed p. 22, the investigation was suggested to me by Mr Greathead’s 
paper, “Analytical Solutions of some problems in Plane Astronomy,” Camb. Math. Jour. 
vol. I. (1839), pp. 182—187, giving the expression of the true anomaly in multiple 
sines of the mean anomaly. I am not aware that this remarkable expression has 
been elsewhere at all noticed except in a paper by Donkin, “ On an application of the 
Calculus of Operations in the transformation of trigonometrical series,” Quart. Math. 
Journal, vol. in. (1860), pp. 1—15 ; see p. 9, et seq. 
5. In a terminology which I have since made use of : 
The Postulandum or Capacity (□) of a curve of the order r is =|r(r + 3) ; 
and the Postulation (V) of the condition that the curve shall pass through k given 
points is in general = k. 
If however the k points are the mn intersections of two given curves of the 
orders m and n respectively, and if r is not less than m or n, and not greater than 
m + n — 3, then the postulation for the passage through the mn points, instead of being 
= mn, is = mn — -J- (m + n — r — 1) (m + n — »— 2). 
Writing y = m + n — r, and 8 — \ (7 — 1) (7 — 2), the theorem may be stated in the 
form, a curve of the order r passing through mn — 8 of the mn points of intersection 
will pass through the remaining 8 points. The method of proof is criticised by 
Bacharach in his paper, “Ueber den Cayley’schen Schnittpunktsatz,” Math. Ann. t. 26 
(1886), pp. 275—299, and he makes what he considers a correction, but which is at any 
rate an important addition to the theorem, viz. if the 8 points lie in a curve of 
the order 7 — 3, then the curve of the order r through the mn — 8 points does not 
of necessity nor in general pass through the 3 points. See my paper “On the 
Intersection of Curves,” Math. Ann. t. xxx. (1887), pp. 85—90. 
6. The formulae in Rodrigues’ paper for the transformation of rectangular coordi 
nates afterwards presented themselves to me in connexion with Quaternions, see 20 ; and 
again in connexion with the theory of skew determinants, see 52.