PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 283 



Ex.4. ^ +a.r + bv + cz = 



rfi « 



— f +d x + h'y^■c! ; = 



— T -^a x-\-h y + c« = 



These equations can be Arritten in the form 



A.x + b y + cz = (i 

 (^x + B'y + c' z = Q 



a:'x+b"y + G"z=0 



By eliminating y and -- we obtain 



{AB' C"-b" c' A- a" cB'-a' 6 C" + a' b" c + a" b c'}j-=0 



or I — ^ +(a + b' + c) — + (al/ + ac" + bc") — j + oS'c" 



Id f^ ' dt ^ ^ a ti 



-b" c' — I +ab" (f-a" c — i + a:'l/c-a'b — - +c^ be" 

 dt^- d (i di^ 



+ a' b" c + 



a" 6 c' \ x=0 



or I — -^ + (a + 6 + c) — + (a5' + a!c" + 6"c'-6c"-aV-a'6) — . 



i dt^ 'dt ^ > ^f\ 



+ ab' d' + ab' d + d' b' c + (^ be" + a' b" c + a" be' \ x = (i 



which is of the same form as Cor. 2, Ex. 3, Class 1, and its integral is therefore 

 known. 



Knowing x, we have by + cz=f{t) by the first of the three equations, 



bd^y d- z 



and -r-T + '^ 7-4=^ W by differentiation, whence, by substituting the values of 



j4 and ^ from the second and third equations, there results a second equation 



between y, z, and t. from these two equations y and z are determined in terms 

 oft 



Ex.5. Given p + a ^ + ba^=p''/ + g ^f + ry 

 dt dt^ ^ dt ^ ati ^ 



Ix J di z ,, ,dy , d^y 



It dt^ ' dt ^ dti ^ 



di 

 di 



These equations may be written 



(d + a d^ + b) x=(p d + gi d^ + r) y 

 (d + a!di + b')x = (p'd + q'di + r')y 



