554: Prof. F. Slate on a Graphical Synthesis 



both typical situations rather naturally under one point of 

 view is the essential nexus made explicit for equations (2) i 

 Reversal of axes with retention of their identity (so to speak) , 

 makes abandonment of the original cycle inevitable. It is 

 plain from what precedes how that mode of reversal would 

 be carried through to any valid equivalent of 



- ( + V = [K^lK + (%iK + (<?!*! )V 3 '] 



+ [(a^iK + CftsyiK + C^Ova'], . (32) 



intentionally emphatic of original direction as given by- 

 equation (1). Similarly, lay down (v/, v/, v/) as original 

 unit-vectors for ( + V), and throw all necessary changes of 

 scale-factor into the second term by compensations in the 

 other term. Reversal now substitutes (v 1? v 2 , v 3 ) and gives 



- ( + V ) = [(rt^V! + (%i)v 2 + (C 1 £ 1 )V 3 ] 



+ [ («2^i)vi + (&2V1) v 2 + (c 2 ^i)v 3 ] . . (33) 



Reference to equations (27) makes it plain that addition 

 of equations (32, 33) has for result the negative of (M). 

 The terms of the comparison inverted lead to (M) itself ; 

 the thought is the same and the end reached. Finally, it is 

 important for the context to remark how (M) can change 

 sign, the order of its factors being untouched. First by 

 reversal of either factor separately ; or secondl} r , by reversal 

 of both factors, and transition to the other circulation-rule 

 as well ; or thirdly, by change of rule alone. Other 

 bearings of the last possibility, on reversal of (Q 2 , Q 2 "), are 

 evident. 



The device in equations (20) is prompted by the an- 

 nounced purpose that prefaces equations (12). But at this 

 point its consequences are available to round out the method 

 and also add some particulars about (M). Whenever (Q 2 ) is 

 parallel to (ABC) by choice or necessity, (r 2 ) loses definite- 

 ness as first denned ; every such case, however, is still 

 tractable by means of a difference of two vectors drawn 

 from (0). Let these be (S l5 S 2 ), on which (ABC) makes 

 intercepts (s ]5 s 2 ) ; the sum-diagonal (Sx + S 2 ) cuts that plane 

 at (pi), while the difference-diagonal taken in the sense that 

 gives its parallel at (0) a real intercept (p 2 )- is (Si — S 2 ). 

 Consistently, within the generalized construction, 



Si^^iSi; S 2 = ?7 2 s 2 ; S l + S 2 =(»i + « 2 )pi; S 1 — S 2 = (n 1 -7i 2 )p^ 



. . . (34) 



