ME. A. CAYLEY ON TSCHIENHAUSEN’S TEANSEOEMATION. 
575 
Upon writing (B, C, D, E)=(a;^ xf^ f)^ the foregoing values of C, 33, <2!5, JT become 
covariants of the quintic {a, h, c, d, e, fXx, y)\ In fact we have 
|C=2(Tab. No. 15), 
i33=2(Tab. No. 18), 
ie= (Tab. No. 13)^(Tab. No. 14)-3(Tab. No. 15)^ 
f- (Tab. No. 13)^(Tab. No. 17)-2(Tab. No. 15)(Tab. No. 18), 
where the Tables referred to are those of my Fifth Memoir on Quantics, Tab, No. 13 
being the quintic itself. This is a further verification of the foregoing result. 
I will conclude by showing how the formula may be applied to the reduction of the 
general quintic equation to Mr. Jeeeaed’s form x ^-\- ax -\- h =(). It was long ago remarked 
by Professor Sylvestee that Tschienhausen’s Transformation, in its original form, gave 
the means of effecting this reduction. In fact, if the transforming equation be 
= a + 
then the equation in y is of the form 
(a 3$, C,3B, (B.fly. 1)^=0, 
where B, C, 33, (2?, f are, in regard to a, |3, y, s, of the degrees 1, 2, 3, 4, 5 respect- 
ively. An d by assuming, say a a linear function of /3, y, S, s, we may make B=0, and 
we have then C, 33, 0, S functions of the degrees 2, 3, 4, 5 respectively of the quan- 
tities jS, y, S, £ : and these can be determined by means of a quadric equation and a cubic 
equation in such manner that C = 0, 33 = 0, in which case the equation in y will be of 
the required form. For considering /3, y, £ as the coordinates of a point in space, the 
equations C = 0, |3 = 0 wiU be the equations of a quadric surface and a cubic surface 
respectively, and if (3, y, £ be the coordinates of a point on the curve of intersection, 
the required conditions will be satisfied. And by combining with the equation of this 
the quadric surface, the equation of any tangent plane thereto (or by the different pro- 
cess which is made use of in the sequel), we may, by means of a quadric equation, find 
a generating line of the quadric surface, and then, by means of a cubic equation, a point 
of intersection of this line with the cubic surface, ^. e. a point the coordinates whereof 
give the required values (3, y, h, s. And similarly for the new form of TscHiEisrHAUSEN’s 
Transformation ; the only difference being that, starting with an equation in y which 
contains the four arbitrary quantities B, C, D, E, we obtain in the first instance an 
equation (1, 0, C, 33, C, 1)®=0 wanting the second term. And then B, C, D, E are 
to be so determined that C = 0, 33=0. 
To proceed with the reduction, I write the foregoing value of C in the form 
jC=( 4ac—4:5^, 6ad— 6dc , 4ae— ihd , af— le XB,C,D,E)^, 
Qad—6bc, 4ae ■3~205d—24(f , qf-l-15de —16cd, 4hf—4ce 
4ae — 4hd, af -\-lbhe —IQcd, 4hf-\-20ce—24d^^ Qcf—Qde 
af~ he, 4hf — 4ce , ^cf— 6de , 4df—4e^ 
MDCCCLXII. 4 1 
