Hugh MacColl.

Symbolic logic and its applications online

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for QRA; and let M 3 denote y c «. 2 <% . 2. tne coefficient of
a 3 . We get

QRA = (M A + M 3 a 3 )(a 1 + a v . 3 + %)
= (M 1 + M 3 )a 1 + M 3 a r>3 ;



140 SYMBOLIC LOGIC [§§ 156, 157

for a 13 = a r and a 9tl = rj (an impossibility). Hence,

there are only two possible cases when n is an odd

number, the case a 1 (that is to say, a>a v which here

means a>l) and the case a r 3 . For the latter, a r -3 ,

we get

OR Lit M, 1 /_ 4, N

jL_ = ?= — J 2a — a n+1

A ira* A 8a' 2 \

For the first case, namely, the case a> 1, we get

QR_ 7^(M 1 + M 8 )

[ A ~ 7w* A

When the integrals in this case are worked out, the result
will be found to be

9? = _L( o» - a«+i Y 2a - a" ) + — ( a^ 1 - a^+i
A 4a 2 \ A / 8a\



1 \2



+ _ ( 2a — a»+ 1

The expression for the chance— —in the case a>l and

the expression for it in the case a < 1 evidently ought to
give the same result when we suppose a=l. This is
easily seen to be the fact; for when we put a=l, each

expression gives - as the value of the chance — — .

8 A

157. The great advantage of this " Calculus of Limits "
is that it is independent of all diagrams, and can therefore
be applied not only to expressions of two or three vari-
ables, but also to expressions of four or several variables.
Graphic methods are often more expeditious when they
only require straight lines or easily traced and well-
known curves ; but graphic methods of finding the limits
of integration are, in general, difficult when there are
three variables, because this involves the perspective
representation of the intersections of curved surfaces.



§157] CALCULUS OF LIMITS 141

When there are four or more variables, graphic methods
cannot be employed at all. For other examples in pro-
bability I may refer the student to my sixth paper in
the Proceedings of the London Mathematical Society (June
10th, 1897), and to recent volumes of Mathematical
Questions and Solutions from the Educational Times. It
may interest some readers to learn that as regards the
problems worked in §§ 155, 150, I submitted my re-
sults to the test of actual experiment, making 100 trials
in each case, and in the latter case taking a = 1 and
7i = 3. The theoretical chances (to two figures) are re-
spectively -56 and -43, while the experiments gave the
close approximations of *53 and - 41 respectively.



THE END



Printed by Ballantyne, Hanson & Co.
Edinburgh &* London



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Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 11 of 11)