REV. T. P. KIRKMAN ON LINEAIl CONSTRUCTIONS. 281 



in that line (or the iV'* tangent line or plane through that 

 point), upon the locus, and this hy the aid of the ruler only. 



By the same method can be found the value of a given 

 ex-local function F Qci yi z^y whether a?, 3/1 and z^ be the co- 

 ordinates of any assigned point, or of any given plane: 

 i. e., it can be determined whether or no {as\ yi z^ satis- 

 fies the equation F (x y z^zizO. 



We shall, in the first place, exhibit our data in a conve- 

 nient form. Any axes and origin being chosen, we can by 

 hypothesis draw Cartesian co-ordinates through oiu" given 

 points. Let us consider, by way of example, the surface of 

 the second order. Putting x^y^ z^ x^ y^ z^, &c., for the co- 

 ordinates of the nine points, 1^ 2, &c., and a^o ya Zq for those 

 of any tenth point, we shall first frame a paradigm of the 

 surface, which may be compendiously represented tlius— 

 :2 ■\;^ x^ . y' . z^ . ocy . yz . zx . X . y . z , 1 zz : 



the terms of this paradigm are* in number 10 . 9 . 8 . 7 . 6 . 

 5.4.3.2, and are all formed from the first term 

 -^ x^ . if . 2^ , xy . yz . zx , X . y . z . 1, 



t 2 3*3 4155 67 8 9 



by permutation of the sub-indices alone ; the sign of any 

 term being determined by the simple rule, that if it is made 

 firom the first by an odd number of transpositions of single 

 pairs of sub-indices, it shall be negative, and positive when 

 that number is even. Thus the terms 



-^ a^ . tf . 2^ . xy , yz , zx , X . y . z. 1 



3 I 2 00 44 65:6 7 » 8 



•— 0^ . y^ . z* , xy . yz . zx , X . y , z . 1 



3 I 8 4400 55 6 79 8 



have the signs which are prefixed to them. The cyclical 

 permutation of an even multiplet involves always a change 

 of sign, this being effected by an odd number of transposi- 

 tions of single pairs, while that of an odd multiplet leaves 

 the sign unchanged. The former of the two last vTitten 

 terms is made fi"om the first by the cyclical permuta- 

 tion of two even multiplets, the quaternion 12 3, and 

 the duad 8 9, either of which permutations alone would 



20 



