256 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
and consequently the cuspidal edges of the reciprocals of curves of the family (176) 
belong to a similar family obtained by altering a into a and f into its conjugate. 
Also the sum of the exponents m for a curve and the cuspidal edge of its reciprocal 
is equal to 2. 
The develojoahle formed by tlie tangents to the new cuspidal edge is 
1 > = if' + .(186); 
and it may he worth while to verify directly that lines of this reciprocal developable 
are reciprocal to the corresponding lines of (179). Also lines in a united plane 
reciprocate into lines through a united point of the conjugate function ; so that we 
can assert that the number of lines of the developable of a curve whose exponent 
is in which lie in a united plane is the number of lines of the developable of a curve 
whose exponent is 2 — m which pass through a united point. 
46. The points (s) in which an osculating plane (184) at (t) cuts the curve again 
are found by combining this equation with (176) and putting 
Spp = 0 = S(/+ if' + = Scf (/+ tf-- (/+ y-a . . (187). 
In this, when we use the expression (178) for n and when we observe (185) that 
a' = [afaf-a] = “ ^ 3 ) (^3 “ fi) (^ “ hO • • • • (188), 
equation (187) becomes 
" 0^(^2 ^ 3 ) ih ~h) {h ~ h) — ^ .... (189). 
Tlie points at which the plane meets the curve four times are determined by 
S(l)-(h + tf{U - -ty){t^-U) = 0 .(190). 
SECTION VIII. 
The Dissection oe a Linear Function. 
Art. 
47. /’= Ff/ -when F = F', = 1. 
48. Condition for tFe reduction/=FR when R = r ( ) . 
49. Reduction of (j to GR when G“= 1. 
50. Reduction of an arbitrary function/= FGR. 
51. Reduction of a function to a product of self-conjugate functions . . 
47 . In addition to the decomposition of a function into its self-conjugate and 
non-conjugate parts by addition and subtraction, there is another very useful 
resolution Iw multiplication and division analogous to Tait’s resolution of a linear 
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