648 REPORT—1899. 
10. An Application and Interpretation of Infinitesimal Transformations. 
By Professor E. O. Lovett. 
1. An infinitesimal transformation is the linear operator represented by the 
symbol 
VPS RIE(ey ery Zn) su . . . e (1), 
if x, . . . %, be regarded as the co-ordinates of a point in space of m dimensions, the 
point (7,,..., %%) is displaced by the transformation (1) to the position (7, + 62, . « », 
x +62), where 
0%, = £08; ... », O&n,=E,0t, - : : : » (2) 
t being an arbitrary differential. The corresponding increment assigned by (1) to 
any function (2, .-.+, Up) is Vor. 
A function is said to be invariant under (1), when its increment due to (1) is 
zero. An equation of any sort whatever w=O is invariant under (1), or said to 
admit of (1) when the increment dw assigned by (1) is zero in virtue of the given 
equation, 
The trajectories of the group of transformations generated by (1) are given by 
the integration of the simultaneous system 
— = =) SREP . . . . 3 
figs fn . 
2. The total differential equation 
, MEP (Gy 00 cp Page, 62 U0, kgs. dagn=0, Str=m,. 9 ai (D 
nm 
homogeneous in the differentials dx, .. . ,, is called a Monge equation ; when the 
Monge equation is linear in dr, ... dx,, that is, of the form 
SIP, «-n.iy Be)Ft= 0,0, sneered Seen 
1 
it is called a Pfaff equation. 
Pfaff equations and Monge equations are integrable or non-integrable, according. 
as certain equations of condition are satisfied or not by the functions P. 
Thus, for example, the Pfaff equation 
Pa, y, z)dv+Q(2, y, z)dyt+R(a,y,2)dz=0  . ; . (6) 
is integrable or non-integrable according as the functions P, Q, R do or do not 
satisfy the well-known relation 
PQ,-R,)+Q(R.—P.)+R@,-G)-0 
By precisely the same method by which (7) is reached as a criterion we find 
the conditions for the integrability! of the Monge equation to be 
PiU. 7 
w 4g S20 ee ee 
0 Ee B eee Sal 
pet A SPE | 84/7 | 
where the Monge equation is 
M,=Pda* + Qdy? + Rdz* + 2Sdydz+2Tdzdx+2Udrdy=0 . . (9) 
1 See Guldberg, ‘Sur la Théorie des Solutions Singuliéres,’ Videnskabsselshabets- 
Skrifter, I. Math.-naturv. Kasse, 1899, No. 4, Christiania. 
