REPORT ON CERTAIN BRANCHES OF ANALYSIS. 293 



The primitive signs 4- and — , when applied to symbols de- 

 noting lines, are only competent to express the relation of lines 

 which are parallel to each other when drawn or estimated in dif- 

 ferent directions; but the more general sign cos fl + ■v^— i sin 3, 

 which has been noticed in the former part of this Report, when 

 applied to such symbols, is competent to express all the rela- 

 tions of position of lines in the same plane with respect to each 

 other. It is the use of this sign which enables us to subject 

 the properties of rectilinear figures to the dominion of algebra : 

 thus, a series of lines represented in magnitude and position by 



«o, (cos 9i + -/ — 1 sin flj) a^, {cos (di + 62)+ V — 1 sin (9i-l- ^2) }«2» 



. . . {cos (fli + ^2 + • • • ^»-i) + ^^^\ sin (51+ 9.2 + . . . 9„_,)} «„_„ 

 will be competent to form a closed figure, if the following equa- 

 tions be satisfied : 



then these fundamental equations will become 



sin & cos ^' surs ^' + sin ^' cos & surs 6 



T ^^'^^'- 1 + e2 c2 sin2 6 sin^ $' ~' 



e e 



cos 8 COS 6' — c2 sin & sin 6' surs 8 surs i' 



f A \^ AW e e e e e e_ 



COS (» + ff; _ 1 +e2c2^2 ^-^ina ^' • 



e e 



surs d surs 6' + e'^ sin i sin & cos & cos d' 

 surs C« + tf ) _ 1 + e2 c2 sin2 ^ sin2 d' 



e e 



If we add, subtract and multiply, the elliptic sines, cosines and sursines of 

 the sum and difference of 6 and 6' respectively, reducing them, when necessary, 

 by the aid of the fundamental relations which exist amongst these three tran- 

 scendents, we shall obtain a series of formulae, some of which are very remark- 

 able, and which degenerate into the ordinary formulae of trigonometry, when 

 c =: and c = 1 : we shall thus likewise be enabled to express sin n 6, cos n 6, 



e e 



surs n 8, in terms of sin 8, cos 6, surs 6. The inverse problem, however, to express 



« e e e 



sin 6, cos i, surs 6, in terms of sin n 6, cos n d, surs n 6, is one of much greater 



e e e e e e 



difficulty, requiring the consideration of equations of high orders, but whose 

 ultimate solution can be made to depend upon that of an equation of (w -|- 1) 

 dimensions only. It is in the discussion of these equations that Abel has dis- 

 played all the resources of his extraordinary genius. 



It would be altogether out of place to enter upon a lengthened statement 

 of the various properties of these elliptic sines, cosines, and sursines ; their 

 periodicity, their limits, their roots, and their extraordinary use in the trans- 

 formation of elliptic functions. My object has been merely to notice the ru- 

 diments of a species of elliptic trigonometry, the cultivation of which, even 

 without the aid of a distinct algorithm, has already contributed so greatly to 

 the enlargement of the domains of analysis. 



