36 Mr. Children's Abstract of the Characters of 



sary to interpolate downwards, and to have regard to the dif- 

 ferent signs. But it is more advantageous and more easy to 

 interpolate from the middle, or from the place where x lies. 

 In all formulae we have not had reference to a definite arrange- 

 ment ; and we may adopt as the first, any other quantity besides 

 P. For the arrangement R, Q, S, P, T the formula (II) becomes 

 X4 = R + {x-r) [r . q\ + [x-r) {x-q)) [r.q.s-] 

 + {x-r) {x-q) [x-s)\_r . q . s .p] 

 or changing the order of the letters : 



X4 = R + {x-r) [9 . r] + {x-r) {x-q) [? . r . s] , jjj. 

 + {x—r) {x — q) {x—s) [^p .q.r .s'] ^ ' 



Comparing this to the above arrangement of the quantities, it 

 will be seen that the difference-quantities here used [gr . r], 

 \jl-r . s], [p . g . ?• . s] are alternately above and below a hori- 

 zontal line, which may be drawn between R and [j . r]. In 

 like manner the arrangement R, S, Q, T, P would give this 

 formula : 



Xi = R + {x-r) \r.s'\ + {x-r) {x-s) [y.r.s] ,jyx 

 -\-{x—r){x — s){x — q)\(i.r.s.t'\ ^ 



where the horizontal line must be drawn between R and [r . s]. 

 Formula (III) is for the case in which x is between q and r. 

 Formula (IV) for the one in which x is between r and s. Both 

 expressions may be comprehended in one, by denoting all 

 quantities which are on one side of x by a^, «,j_i, •••••• «i (^'^ 



among which let a be nearest to x ; and those on the other side, 

 by l),b^...b_^b^ where b is supposed nearest to x. Sup- 

 posing now that a is always the quantity nearest to j", and b 

 more distant, both formulae will become as follows : 

 X = A + {x—a)la.b~\ + {x-a) {x-b) [a.ab'] 

 + {x—a) {x — b) {x—a,) [a, abb,] 

 + {x— a) {x — b) (x — a,) {x — b,) [a^a, abb,}.... (V) 



[To be continued.] 



V. An Abstract of the Characters of Ochsenheimer's Genera 

 of the Lepidoptera of Europe; mth a List of the Species of 

 each Genus, and 'Reference to one or more of their respective 

 Icones. By J. G. Children, F.R.S. L. Sf E. F.L.S. Sfc. 

 [Continued from page 450.] 



Genus 24. MACROGLOSSA, Ochs., Steph. 

 Macroglossum, Scop. Sesia, Fab., Syst., Gloss. 



BoMBYLi^, Hlibn. Sphinx, Ochs. 



Wings, small in proportion to the body ; (elongate-lanceolate, 

 opaque. Steph.) 



Antenna; 



