of Edinburgh, Session 1883-84. 
299 
obtain the curves shown in fig. 4 ; symmetrically placed in regard 
to the origin 0 ; the one curve being converse to the other. 
We may put these formulae in a more convenient form by trans- 
posing the origin of abscissae in the one case to - Jtt, in the other 
to + Jtt ; for this purpose we write x — ^ir + i ov —x = ^7r-i, and 
in this way confine our attention to one of the curves, whose equa- 
tion now becomes 
l = xcosx- sina;. 
In order to determine the singular points in the curve, we must 
take the first and second derivatives (differential coefficients) of 
this expression; these are 
SI 
or iJ = x.sina;, 
Sx ’ 
sn 
^ or = -\-x.ao?ix + sin x ; 
and continuing the derivation, 
or 3 ^Z= -fl?.sinar-f 2cosa* 
hx^ 
or J = 
- xaoBx - 3sina; , 
etc. 
etc. 
The remarkable simplicity of this progression suggests the con- 
tinuation in the opposite direction ; taking the successive primitives 
(integrals), we get 
fSx. I on = - a^.sina? — 2cosaj, 
ffSx^.l or +x.Q,OBx- 3sina?, 
fffSx^Aov -f- a?, since -1- 4cosiu , 
and so on ; 
and it is obvious that, in this endless progression of derived 
functions, the appropriate middle term is ce. since, which also has its 
conjugate a?.cosce. Denoting the one of these functions by the 
other by we have the conjugate progressions : — 
