(2) 



457 



Ml = o'm'i + bH'i + c^w's + be {v'3 + w'a) + ac (iv\ + u'3) | 



+ ab {u'i + v'l) 

 V2 = d'^u'i + b'-v'-i + c^w'z + h'c (v'3 + 2^2) + a'c' (w'l + u'3) 



+ db' (tt'2 + v'\) 



t^3=a"2M'i + i"=^u'2 + C"'^W'3 + 6"c" (y'3 + W'i) 



+ a"c" (it--'! + M'3) + a"6" (w'a + v'l) 

 V3 + t^2= 2a'aVi + Ibh'v.i + Iccw':^ + (i'c" + 6V) (w'a + w'2) 



+ {ca + cV) (m;', + M'3) + (a'6" + d'b) (^'2 + u'l) 

 if;i + M3= 2aa'Vi + 266"y'2 + lccw'3 + (6"c + 6c") (w'3 + w'.^ + 



(c"a + cd) {w\ + m's) + {db + a6") (m'j + v'\) 

 M; + Ui = 2aa'M'i + 266'y'2 + 2cc'w'^ + {be + b'c) {v'3 + w'^) + 

 {cd + cd) {v}'\ + M'3) + {ah + db) {u^ + v'\). 



These equations of transformation are identical with those for 

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



Theorem II. 



" Let X, /u, V, ^, x» ^ be functions defined by the follow- 

 ing equations : 



\ = ai^ + /3i* + 71^ ^ = 0203 + /SijSa + 7273 



H = Oi + j32- + 72^ X =" °3ai + /Bs/Bi + 7371 



V = 03^ + jSa^ + 73^ i/* = aia2 + ^i/32 + 7172 



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

 into the following linear functions of similar quantities : 



A = a^X + ^-V' + c^v' + 2bc 0' + 2cax' + 2a&i//' 



X = d^X + i'V + c'^v + 26'c>' + 2c'a'x = 2a'J'f 



V = d'-X + 6"V + c"'^v' + 2b"c"<^ + 2c"a"x' + 2a"i"^' 



^ = a'a"X' + b'U'fx + cV'v' + (^''c" + b"c') ^' + (c'a" + c"d) x 



+ {db" + d'b') xp' \ (3) 



X = ad'X + bb"fi + cc'v -^ (6"c + be") ^' + (c"a + cd') x 



+ (a"6 + ab") i//' 

 i/» = aa'X' + bb'ix + cc'v' + {be + i'c) 0' + {cd + c'a) x' 

 + {ab' + a't) »/-' 

 2 H 2 



