TRANSACTIONS OF SECTION A. 44.9 
é.e., a function such that the unknown function x(s) is given by the formula 
x)= +] KG OMOat. eras =, (10) 
the double integral bb 
Q a=| | K(s, t)f(s)f(é)dsdt . ; : Teens) 
increases with A, becoming infinite and changing sign only as \ passes through a 
singular value for equation (8). When \= — o, it can be shown that Q(A) =0, and 
when A =0, K(s, t) reduces to «(s,¢). Hence if there are no negative singular 
values of A, Q(A) is positive for A =0, ze., 
b fb 
| |. o, oneyrawt 
as a 
is positive ; also f(s) is an arbitrary continuous function, and so x(s, ¢) is definite. 
§ 4. The double integral (5) being considered as analogous to a quadratic 
form, that arising from a definite function corresponds with a quadratic form which 
is essentially positive. There is, however, a more general expression, viz., 
b bb 
| (ots) asf | k(s, t)w(s)o(t)dsdt f : (12) 
a aja 
which also corresponds with this, aud we can show that if the function «(s, ¢) is such 
that this expression is positive for every continuous function o(s), the singular 
values of d for equation (8) cannot lie between 0 and p. 
When x(s, ¢) is not a symmetrical function of s and ¢ the singular values of 
A are not necessarily real, as is the case when x(s,¢) is, symmetrical. Definite 
information with regard to their nature can be obtained, however, in the following 
cases, the first of which is mentioned by A. Myller, and is an extension of a 
theorem in quadratic forms due to Weierstrass :— 
(1) If x(s, t) = —x(¢, s), the singular values of \ are purely imaginary quantities, 
(2) If 
K(e, t) -| fle, wale, t)de, 
a 
where /(s, x) and g(a, ¢) are real symmetrical functions of their arguments and 
one of them is definite, the singular values of ) are all real; if both functions are 
definite the singular values of A are all positive. 
5, Operational Invariants. By Major P. A. MacManon, F.R.S. 
6. A Method of obtaining the Principal Properties of the Exponential 
Function. By Professor A. E. H. Love, /. 2.8. 
It is desirable to arrange the theory of the exponential function in a form 
which shall be at once simple, rigorous, and systematic. In order to attach the 
theory to familiar things, we may begin by attempting to differentiate log,,.«; for 
computation with logarithms, the appearance of the logarithmic curve, and the 
method of differentiation, as applied to simple rational functions, ought to be 
familiar to students before they proceed to the exponential function, Since 
Ah { log, (+h) —logy, x } =a log, (1 + Ajax)", 
the process of differentiation naturally introduces the limit 
limpow (1+1/n)", 
1907. GG 
