TRANSACTIONS OF THE SECTIONS. 3 
we shall have the expression 
ive) . nN oth on 
pai’ ip (AR yo SWB by 9: dice lon on hy 
eS ~ 
as a transformation of the formula 
1 n n(n—1 
Pe ton Fey OS ety" = Ke, 
hess (10.) 
—(n—c) + - (n—e—2)” = is )) (n—c—4)"+ &e. }s 

each partial series being continued only as far as the quantities raised to the nth power 
are positive. Laplace has arrived at an equivalent transformation, but by a much less 
simple analysis. 

On some investigations connected with the Calculus of Probabilities. 
By Sir Witt1am Rowan Hamitton. 
Many questions in the mathematical theory of probabilities conduct to approxi- 
mate expressions of the form 
2 Pipi ba 
P=anf die™™, that is, p = © (2), 
0 
© being the characteristic of a certain function which has been tabulated by Encke in 
a memoir on the method of least squares, translated in vol. ii, part 7. of Taylor’s 
_ Seientific Memoirs, p being the probability sought, and ¢ an auxiliary variable. Sir 
William Hamilton proposes to treat the equation p = © (4) as being, in all cases, ri- 
_ gorous, by suitably determining the auxiliary variable ¢, which variable he proposes to 
call the argument of probability, because it is the argument with which Encke’s Table 
_ should be entered, in order to obtain, from that table, the numerical value of the pro: 
bability », He shows how to improve several of Laplace’s approximate expressions 
for this argument ¢, and uses in many such questions a transformation of a certain 
double definite integral of the form = - 
1 . ao 2 1 
a dr due’ Ucos (2 rs’ uV)= Or+tyirs Jide Wyrs A, 
0 0 
in which 
U=1 + aU? + au... ; V=1 + 6, w+ Bu’. ” 
while », v.... depend on #;.,.,;..and7; thus »} = 4¢:— 17°. The function © 
has the same form as before, so that if, for sufficiently large values of the number s 
(which represents, in many questions, the number of observations or events to be 
combined) a probability p can be expressed, exactly or nearly, by the foregoing double 
_ definite integral, then the argument t, of this probability p, will be expressed nearly 
_ by the formula 
Re ye ea fey 
t=r(l dein * 4. V2 2), 
Numerical examples were given, in which the approximations thus obtained ap- 
ared to be very close. For instance, if a common die (supposed to be perfectly fair) 
_ be thrown six times, the probability that the sum of the six numbers which turn up in 
_ these six throws shall not be less than 18, nor more than 24, as represented rigorously 
_ by the integral 
a 2 77, sin7a /sin6x\° . . 27448 
= 5/2 aa (; aa) , or by the fraction Tax i 
while the approximate formula, deduced by the foregoing method, gives 27449 for the 
numerator of this fraction, or for the product 6° p; the error of the resulting proba- 



