of Edinhurgli, Session 1883-84. 
605 
mate quotient unit. 
The fallacy of these arguments may he clearly seen from another 
point of view. A certain curve crosses the line of abscissae at equal 
intervals extending indefinitely both ways. If we place the origin 
of the abscissae at the middle of one of the intervals, and denote the 
ordinate y by the general symbol (fiX, y becomes zero on substituting 
for X any odd multiple of the half interval ; which half interval we 
shall denote by q ; that is to say, , <^( ± 3q) , ± 5q) , &c., 
are all zero. Hence, according to these arguments, the expression 
for y or (f>x must be the continued product of the factors 
and thus our curve can be none other than the curve of sines ; or, 
to state the conclusion in all its absurdity, no curve other than the 
curve of sines can cross its axis at equal intervals extended inde- 
finitely both ways ! 
of all the coefficients a, (3, y ; that of is the sum of the 
products of each pair of them ; and so on, or, in the usual notation, 
(1 - ax^)(l - Px^){l - yx^~) . . . = 1 - 4 - x^ta(3 - x^ta^y . . . , 
and thus, in accepting the above arguments, we assume, inad- 
vertently, that the interminate series 
amounts exactly to \ ; that is to say, we assume the equality 
In the continued product of any number of factors of the form 
1 - 1 - (3x^, 1 - yx"^ , the coefficient of x^ is the sum 
1 1 1 
^2 +952 2523’^ ■■■ 
or 
and have accomplished the very difficult problem of summing the 
