192 REPORT—185/7. 
It would appear that problems of this sort are not soluble by algebra alone, 
when the problem itself contains an indeterminate element; but if it contain 
an infinite in lieu of an indeterminate element, it is soluble by algebra, as a 
limiting case of some more general problem. Suppose a person plays a 
succession of games, of which, when divided into sets of three, he wins in 
every first game @ pounds, in every second b pounds, and loses in every third 
game a+ pounds; what is the present value (apart from any considerations 
of time) of what he will gain, supposing that he plays until the happening of 
some event unconnected with the game and which may never happen? The 
answer given by the Theory of Probabilities is} (2a+6), being the mean of 
the three possible results of one set of games. If the conclusion of the play- 
ing be indeterminate, algebra gives only an indeterminate result ; but if the 
conclusion of the playing be indefinitely postponed, the answer to the problem 
is, the value to which the series az + bx* + cx*+ ax‘+ bx’ +cx'+ ..... (¢ being 
—(a+b)) approximates as @ approximates to unity; and this also is 
4(2a+48). The problem is in fact brought within the domain of algebra by 
considering it as the limiting case of another problem, in which an algebraic 
relation exists between the values of the winnings of each game; and the value 
of the limit depends upon the nature of the connexion. In the problem pro- 
posed, the @ pounds won in the first game has no kind of connexion with 
the @ pounds won in the 4th, 7th, &c. games; but in the extended problem 
a connexion exists, inasmuch as we consider that when az is the value of 
a pounds at the end of the first game, az*, az’... represent the values of 
a pounds at the end of the 4th, 7th, &c. games; and it is this consideration 
which renders the problem an algebraical one. 
Considerations of this nature seem to tend to the conclusion, that we are 
not to expect from algebra alone, as that science is at present constituted, the 
discovery or proof of the principle that the analytical equivalent of an inde- 
terminate expression is the arithmetical mean of all its values, or any prin- 
ciple of this character; for algebra being the science of symbols irrespective 
of their meanings, knows nothing of symbols which are of their own nature 
periodic or alternating, or otherwise limited as to the values they are capable 
of having, or of symbols which are indeterminate in point of value; except 
indeed when they occur as limiting cases of particular problems; in which 
cases their values are to be learnt from the specialties of the problem, or in 
other words, the science to which the data of the problem belong. 
Returning to the consideration of infinite angles, it will be readily seen that 
Oc, and O,_, are identical in every particular except sign. Considering 
these angles as longitudes attained, they possess a species of identity which 
OC, and OC,, do not possess ; the former pair have passed through precisely 
the same values, have traversed every meridian the same number of times ; 
the latter pair have not. Viewing the two pairs of infinities as angles im- 
pressed with every value through which they have passed, now that they have ~ 
arrived at an indifferent or neutral value, the former pair possess an absolute 
identity (except in sign) with reference to our mode of interpretation. If we 
take two infinities which differ otherwise than in sign, and if we permit our- 
selves to say that (—1)*=0 and (—1)*’=0, yet we could not thence infer 
that (—1)*=(—1)®’ ; but since —1=—1~' identically, we may be sure 
that (—1)“g=(—1)°%7-8 since &,=—QC__,; and if these quantities are to 
be interpreted each into zero, we may be sure that it is the same zero. But 
whether it be, or be not, safe to affirm that (—1)~=0 universally, we may cer- 
CHE I 
inl infinities being tl »,) that ——~___* _$__=—, 
tainly affirm, (the three infinities being the same,) tha (=1)84(—=1n 
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