136 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



press, not the cosecant and cotangent, but the inverse functions corresponding to sine 

 and cosine, or the arcs which are more commonly called arc (sin = s), arc (tan = t), 



dr 

 It must also be observed that the factor ^^/^j which we have introduced under the 



signs of integration, is not superfluous, but is designed to be taken as equal to posi- 

 tive or negative unity, according as c? r is positive or negative ; that is, according as 

 r is increasing or diminishing, so as to make the element under each integral sign 

 constantly positive. In general, it appears to be a useful rule, though not always 

 followed by analysts, to employ the real radical symbol ^R only for positive quan- 



T 



titles, unless the negative sign be expressly prefixed ; and then -j= will denote posi- 

 tive or negative unity, according as r is positive or negative. The arc given by its 

 sine, in the expression of the element a, is supposed to be so chosen as to increase 

 continually with the time. 



35. After these remarks on the notation, let us apply the formula (P^.) to calculate 

 the values of the 15 combinations such as {», ?i},of the 6 constants or elements (Q^.)- 



Since 



r= v(i2 + ^2 + ^2), (187.) 



it is easy to perceive that the six combinations of the 4 first elements are as follows : 

 {k,\} = 0, {k,il} = 0, {k,v} =0, {\l^} = ^, {X,v} = l, {;a.,^} = 0. . (188.) 

 To form the 4 combinations of these 4 first elements with r, we may observe, that 

 this 5th element r, as expressed in (Q^.), involves explicitly (besides the time) the 

 distance r, and the two elements x, (Jj ; but the combinations already determined 

 show that these two elements may be treated as constant in forming the four combi- 

 nations now sought ; we need only attend, therefore, to the variation of r, and if we 

 interpret by the rule (P^.) the symbols {», r} {X,r} {[Jij,r} {v,r}, and attend to the 

 equations (P.), we see that 



dT 



{;s,r} = 0, {X,r} = 0, {|U.,r} = -^^, {v,r} = 0, (189.) 



dv 



-^^ being the total differential coefficient of r in the undisturbed motion, as determined 



by the equations (I^.) ; and, therefore, that 



{«,r} = 0, {X,r}=0, {»',r}=:0, (190.) 



and 



r 1 ^T dr dtdr rim \ 



i(^>''} = -fT-d-t= + Trrt = ^'' (1^1-), 



observing that in differentiating the expressions of the elements (Q^.), we may treat 

 those elements as constant, if we change the differentials of | ;j ^ a?' 3/' z' to their undis- 

 turbed values. It remains to calculate the 5 combinations of these 5 elements with 

 the last element a ; which is given by (Q^.) as a function of the distance r, the coor- 

 dinate ^, and the 4 elements «, \{J(*, v; so that we may employ this formula, 



{^.«}=TFK^} + |f{^,0 + |^{e,«}+|^{e,X}+|^Kit.}+|^{e,.}, (192.) 



