;i 



HENSLOW ON THE VARIATIONS IN 



la endeavouring to trace the transitions which so frequently occurred in the stems 

 from one kind of divergence to the other, I met with some which were new to me 

 Thus $ wa-, as given above, by no means uncommon; another was-^; and occasionallv 

 an approximation to J and -^ appeared. Now these fractions can manifestly be 



d into a scries analogous to the usual 



12 3 5 



4J 7> 1 1J 18 



&C 



v\ here we see at once, by comparing them with 



3> 5J 8J 1 3> °^-> 



tli ! while they retain the same numerators, the denominators of the successive fractions 

 of this secondary series consist of the sum of the numerators and denominators of the 

 corresponding fractions of the first or primary series, as I propose to call it. 



I must here observe that the primary series arises from a fact which appears to be a 

 very important one, and, as far as I am aware, has not hitherto been noticed, i.e. that 

 every coil of the helix commencing with the position of any leaf, contains two other 

 leaves beside the initial one. Thus, referring to fig. 1, if we start from leaf JS T o. 1, No. 4 

 is Dot reached until after the coil has been completed, on arriving at the vertical line 

 through the initial leaf; similarly, commencing with No. 4, No. 7 is found to be in the 

 next coil ; so that no such coil ever contains more than three leaves. 



In the secondary scries every such coil will contain four leaves. The same numerators 



ned for the corresponding fractions of both series, as they represent the number 

 aeli cycle, and they are not altered by the number of leaves being increased 



of coils 



in each 



i 



In a similar way it may be seen that tertiary, quaternary, and other series might he 

 (luccd. in which each coil will contain five, six, or more leaves successively. And, 

 farther, the following algebraical expressions will represent every series of divergence. 

 Let a be any number ; then the fractions 



112 3 5 8 



a 



« + l 2a+l 3« + 2 5a + 3 87+5 ' &C ' 



arc s quite general and will include the angular divergences of all generating spirals. 

 They are based upon the principle that any portion of the spiral subtending 360°, i. e. 



:- smgk coil of the helix as above described, contains «+l leaves. To each series there 



IZZ r H lraC,IO, \ V1 1 1 -; t0 thG V*™* i t0 the secondary fc to the tertiary h Ac., 

 - st be exempted from the foregoing remarks, as the leaves in a projected coil of 



41 . . v ulLS>e initial tractions are flip pnrmonf;™ i^v* iwnraoi 



the several systems or series of divergent™ * 



the connecting links befrw 



* bj coring the existence of other series than 



thence d bang the algebraical formula in ~~— * .„ 



and 



found, bv referring to MM. Martin 



- »"* of MM. Schimner and TXr T tUVer ^ ences ' I h ave found, by referrim 



- . the same , ^ r l^^T S ***** - Ataxia (Ann. des Sc Nat. 2* 



same expressions have already been surest.,! * ■ ! lAnn * aeS Sc ' * at - *°* 8 ^ ™ ]837 >' tnat 



'1 that a single species has so manv ZiZ * independent 80uree8 - It is, however, I think, interest- 

 t i rr eonauWnKi ,. .. 7 vanaUons oi arrangement that it alone bns *ffnr^ th» means of 



m ng at very considerable generalizations. 



