457 



K2) 



Ml = c^ux + hH<^i + c^w\ + he (v's + w\) + ac (w\ + m's) 



+ ab (w'a + w'l) 

 ^2 = a'^u'i + b'H\2 + c"^w'z + ^V (v'3 + w'^) + dc (w\ + u'z) 



+ a'b' {u.2 + v'l) 

 Wz=a'^U\ + *"^«'2 + c'^wz + 6"c" (y'3 + w'2) 

 + a"c" (?^-'i + m's) + «"&" (^2 + «'i) 

 ^3+^2= 2a'a"M'i + 1hh'v<i + ^cd'w'^ + (6'c" + 6V) (w'3 + m7'2) 



+ {c'a + cV) (tf?', + w'a) + (a'6" + a"6') {u^ + ^'i) 

 M7i+M3= 2aaVi + lbb'v'2 + 2cc"w'i + {b"c + 6c") (v'3 + w'.^ + 



{c'a + ca") {w\ + m's) + («"6 + ah') {u^ + ^?'l) 

 M2 + i;i = 2aa'M'i + 266 V2 + Iccw'^ + (6c' + 6'c) (v'3 + w'2) + 

 (ca + ca) {w'\ + 2/3) + (a6' + a'6) (^'2 + v\). 



These equations of transformation are identical with those for 

 the transformation of x^, y^, z^, 2yz, 2xz, 2xy. 



Theorem II. 



" Let X, ju, V, 0, X' ^ be functions defined by the follow- 

 ing equations : 



\ = ai^ + jSi^ + 71^ = 0203 + ^2/33 + 7373 



ju = a2^ + jSa^ + 72^ X = "3ai + /Bs/Bi + 7371 



V = as^ + fis^ + 73^ ^ = aia2 + /3i/32 + 7172 



These functions will be changed, by a change of co-ordinates, 

 into the following linear functions of similar quantities : 



X = a^X + 6V + c^v + 26c (^' + 2cax' + 2a6f 



X = a'^X + 6'V' + c'^v + 2b'c'(p' + 2c'a'x' = 2a'6'^' 



V = d'^X + h'-^fj! + c'^v + 26"c"^' + 2c"a"x' + 2a"6"^' 



^ = dd'X + 6'6V + ccv + (6c" + 6"c') ^ + {cd + cd) x' 



+ (a'6" + d'h') xp' \ (3) 



X = ad'X + 66"/x' + cc'v -r {b"c + be") ^' + {c'a + ed') x 



+ {d'h + a6") -^ 

 ■^ = aa'X + 66y + ee'v + {be + b'c) <^' + {ed + c'a) x' 



+ (a6' + db) \p' 

 2 m2 



