2 REPORT—1845. 
On Triplets. By the Rev. C. Graves, M.R.J.A., Professor of Mathematics 
in the University of Dublin. 
An abstract of this communication will be found in the ‘ Proceedings of the Royal 
Trish Academy,’ vol. iii. parti. 
On the Equation of Laplace’s Functions. By Grorce Boo.e. 
The method by which the integral was obtained is the one developed in a paper 
on a General Method in Analysis, published in the Philosophical Transactions for 
the past year, and I may be allowed to express my opinion, founded on a very care- 
ful examination of the subject, in which I have obtained no less than six distinct 
forms of the integral, that this is the only way in which we can obtain asymmetrical 
solution free from integral signs. 
The equation of Laplace is 
a URS § ERAT ee ny, bs 
du & dp ioe tt et De=s 
and the integral in question 
eer ( oe) 
& 
where 
d 1\a ov-1 ov-1 
f— aFndlt és 
F(é yao {q@tey'y ihn ) + enn —; )}. 
w and x denoting arbitrary functions. This integral consists of two particular ones, 
and I have proved that when the arbitrary functions which they involve are so as- 
sumed as to produce together real forms for ~, the results are the same as would be 
got by confining ourselves to one solution, and equating separately to 0 the real and 
the imaginary portions. This seems to be an usual, perhaps it is a general property of 
differential equations which involve in their solution the imaginary unit ./— 1; and 
it is important, because it shows that a particular integral taken with this license of 
interpretation becomes a general one. 
By giving to the arbitrary functions the general form ¢ ¢<” @¥—-11 ote OV—-1 ond 
prefixing = to the result, I have obtained the following real form of solution, 
u= {af (u)+bf (—#)} cosrQ; 
with a similar one in multiple sines, a and 0 being arbitrary, and 
dl be _ 
f@ == any a y eer ty)” r. 
The summation is quite unlimited with respect to r, that is, » may be positive or 
negative, integral or fractional. In all cases the coefficient of cos r@ is a finite 
function of ~, which is an unexpected result. 
In the case of Laplace’s functions, the summation extends from 0 to 2, including 
only integer values of r. On determining the constants, we get’ 
P =A,+ 2 (Aicos? + A,cos2 @..+ Ancosn 9), 
where in general 
f (w)F (H) 
~2..8r 1.2.3.8 
and f («) is as above. I think it is evident that in deducing these functions from 
the general integral of the equation which they satisfy, we not only employ the na-— 
tural and direct method of analysis, but also gain something in the final result, as 
respects simplicity of form and elegance of expression. 
Ar;= 1 
On the Premises of Geometry. By H. Wepcawoon, M.A. 
The conception of geometrical figure, as traced out by the motion of a point, ne- 
cessarily supposes in the student the capacity of comparing the direction of that mo-— 
