160 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
“ bs ~ ^rs^4 ^r^4S ^s^ir "1“ \^rs ^r^\s ^^yrs 
+ fn^sVl “t" ^2sVt ^2'H'Vs ^rVis "t" ”1" ^2Vts ^^r'^is ‘^Virs 
^rsCi + 9rs^i + ^3S^r + ^s ^r^4s ffs^-ir + ^sCrs + + ^Cirs + 9^\rs 
+ d^gd^ 4* ^ ”^' ^r^is ”1” ^s^4r “f" rs ^r^\s ^s^ir \rs "^" ^^ 4 rs] > 
7)lj. = e[Zy^2 “H ^r^4 "t" ^^2^ 
+ m/r)^ + b^Tj^ + m^r]r + mrj^r + ^Vir 
^r^2 rCi + ^3^r '^^2'f ~^f^4:r 
+ d^6^ + ?>^r^4 + ^/ir^4 + ^^2r + ^^^4r] j 
^^rs = ^[^rsf2 + ^rs^4 ^hr^s + + ^r^4s + ^s^4r + h^2r + h^2s + '^h^rs + ^^2rs + ^iirs 
+ ^rs^2 ^rs9i ^^^2rVs ”1" '^2sVr ^hVir ^^s 9‘1T '^r'92S '^‘d^rs '^'92fs ^^4rs 
+ '^rs^2 "l".Z^rs^4 + '^ss^r + frC^s +./s^4r + '^s^ 2T ^r^2s ^^2»’s fC^rs 
+ dj-sO^ + + mr^^s + ^^s^4r + ^s^2r + ^r^2s + ^4^rs + <^^2rs + ^^4rs] 
- 7^^ = e[/|.^3 + + /^3r + ^^4r 
+ ^^hVB +frV4 + ^2Vr + '^^Vsr +fVir 
+ Wy^3 + Cj.^^ + 773 ^t- + 7^^3t- + 
+ ^r^3 "^' ^^r^4 “I" ^^4^r + dd^^ 4- , 
■ ^^rs — ^\}rs^z drs^4 + ^]r^s + ^is^r + 9r^4s 9 s^4T + ^s^sr + ^r^3s + ^i^rs "I" ^^3^s “f“ 9^4rs 
+ m^giq^ +frs'94 + ^2rVs + '^2sVr +frVis +fsV4r + ^hlsr + + ^2^rs + '^^Vbts +fVirs 
■^" ^rs^4 ^3s^r ^r^4s “^" ^s^4r "t" ^s^b'^' ^^rCzs “I" ^^s^rs "I" ^^srs "1" ^^4rs 
+ dj.gO^ + Tlj-gO ^ + yi^fO s 4“ '^ij^s^r ~f” 4s ^s^4r ^^r^Bs ”1” ^^3rs ~i“ ^^4rsJ > 
for ?",« = ], 2, 3, 4 in all combinations.* 
15. Thus, within the range of derivation retained, we have, for each of 
the ten coefficients in the quadratic form, the coefficient itself, its four first 
derivatives, and its ten second derivatives ; consequently there are 150 
quantities arising through those coefficients. For the concomitants which 
are to be associated with the form in the complete system there are the 
four variables X, the six variables P, and the four variables U ; that is, 
fourteen more. Hence the total number of quantities that occur is 164. 
As regards the derivatives of each of the quantities rj, 6, we have 
the four of the first order, the ten of the second order, and the twenty of 
the third order, while the quantities r], 9 do not themselves occur in 
the infinitesimal variations of any of the 164 quantities occurring. Thus 
the total number of derivatives that have to be taken into account is 136. 
It is convenient to arrange them in sub-groups of 80 (for the third order 
of derivation), 40 (for the second order), and 16 (for the first order). 
* These ten sets of symbols can be grouped into a single set, giving 
/ dgpq \ dgpq ( d‘^9pq \ b^9pq 
\ dx^ ) dXr ’ \dXrdx,J dx^dxj 
for all values of p, q, r, s ; but the forms are not so easy to manipulate as are the more 
diffuse expressions in the text. 
