118 Professor Hamitton’s Third Supplement 
sion which leads to the received phrase of least action, a maximum or minimum of 
the function V+ VY; but in order that this value should really be greater than all 
the neighbouring values, or less than all, a new condition is necessary. To find this 
new condition, we oh observe that the relations 
eid lle dal al Soi atone 
* 3 = + sca ee = ~ (4 get Bart Veal) > 
BF 8 eros ALA LA a 
® sy t Poy Se re eae andy + 8 o T eayse ‘)s (G") 
payee OV Se OF eV, gals 
* Srdz t Baraat ae 737 (4 ovdz + BS a 822 ay J 
which result from the third number, give 
eV SY, as \?, (VP, Seen. 
8 +80 = (+ ) (se —£ as ) = Soe: 82) 
5 ay ana Be nN, " 
+2 (seay t gaa) (82-5 = “ az) (8 mp 2) 5 (H") 
the condition of existence of a maximum or minimum, properly so called, of the 
Function V + V,, is therefore, 
20 dhe = Get a) yw) (et mE) a 
When we have on the contrary 
Q<0, (K") 
the variation of the second order ®V+&V, admits of changing sign, in passing from 
one set of values of dx, dy, dz to another, that is, in passing from one near point B’ 
to another ; and since, to the accuracy of the second order, 
V (A, B’) + V (B, C)-—V (A, C)=4 (8V+EV), (L”) 
we shall have the one or the other of the two following opposite inequalities 
V (A, B’)+ V (B, C)> or < V (4, C), (M") 
according as the near point B’ is in one or the other pair of opposite diedrate angles 
formed by two separating planes P’ P” determined by the following equation 
SV+8V,=0, (N"”) 
which is, by (#7"’), quadratic with respect to the ratio 
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