NA TURE 



[November 7, 1901 



followed into the proximal end of the spinal coid. In 

 ornithorhynchus, Ziehen could not satisfy himself about 

 the existence of any pyramidal system. Prof Kcilliker 

 believes he can distinguish a pyramidal decussation in 

 ornithorhynchus and that the fibres of it plunge mainly 

 into the dorsal column of the spinal cord fas in the rat 

 and guinea-pig) and not into the lateral column, as in 

 the generality of mammals. But the description he 

 gives is a very unsatisfactory one, and no mention is 

 made of his data for discrimination between the un- 

 doubtedly existent fillet decussation and the equivocally 

 existent pyramidal. Moreover, he admits that he finds 

 in ornithorhynchus no trace of longitudinal fibres passing 

 anywhere along the pons. In arguing for the existence 

 of a pyramidal system, he omits mention of what to 

 most minds forms the strongest argument yet adducible, 

 namely, that, as shown [by Prof. C. J. Martin, of Mel- 

 bourne, excitation of a certain region of the cerebral 

 cortex of ornithorhynchus evokes movement of the 

 crossed fore-limb. 



The large extent and specially sentient character of 

 oral-facial surface in ornithorhynchus prepares the 

 observer for a large recipient nucleus in the bulb to 

 subserve the huge sensory root of the trigeminus. This 

 root and its recipient nucleus form a huge projection 

 either side the bulb — the tuberculum quinti, well shown 

 in a figure reproduced from Elliot Smith. Prof KoUiker 

 finds fibres of this root traceable to the nuclei of the 

 hypoglossus, vago-glossopharyngeus and abducens, as well 

 as to that of the trigeminus itself. From the recipient 

 nucleus of trigeminus he traces fibres to the median 

 fillet, and so to the optic thalamus. C. S. S. 



DIVERGENT SERIES. 

 Lemons sitr Ics Siries Dh'crgeiifes. Par Emile Borel. 

 Pp. viii-l-184. (Paris: Gauthier-Villars, 1901.) Price 

 fr. 4.50. 



TO make the object of this work intelligible, it is 

 necessary to recall a few facts concerning infinite 

 series in general. Suppose we have a sequence 



K|, »„, U^, . . . ll„, ... (U) 



where Ki, ti-^, &c., are analytical expressions constructed 

 by a definite rule. Let J,i = «^,-f «.,-(- . . . +u„; then we 

 have a derived analytical sequence 



X,, s„, J-,,, . . . s, ; (S) 



this is a definite analytical entity, and its properties are 

 implicitly fixed by those of the former sequence. The 

 expressions «n, s„ are, of course, functions of n ; we may 

 suppose, for simplicity, that, besides this, they involve, 

 in addition to definite numerical constants, a single 

 analytical variable, .r. If we assign to .r a numerical 

 value, S becomes an arithmetical sequence, and three 

 principal cases arise, according to the behaviour of s„ 

 when n increases indefinitely. If s,i converges to a 

 definite limit s we say that this is the sum of the series 



«i-t-K2 + "3+ • • • , and write i = 2//„; but the ultimate 



1 

 value of s may be either indeterminate or infinite. In 

 the second case 5«„ has no definite meaning ; in the 

 third we may say, if we like, that 2/c, is infinite, but this 

 NO. 167 I, VOL. 65] 



infinite sum is not a quantity with which we can operate, - 

 and presents no special interest. 



When the series S/c. and 2?',, are absolutely convergent 

 we can add and multiply them according to the rules 



2ti„ + %v„ = -S,{u„ + v„) 



(A) 

 (B) 



now the sequences 



«j-hZ/„ ll„+V„, . . .. tti,.+ V,„ . . . 

 !/,t',, U^Vn + UtV,, . . . UiV„ + U2V„^^+ . . . +U„Vj, . 



can be constructed, whether or not the sequences 

 (;/„ //o, . . .) and (■:\, ''.,, . . .) are convergent ; the ques- 

 tion therefore arises whether it is possible, even when 

 the series 'Su,,, 2v„ are divergent, to associate with the 

 sequences (w,, «._> . . .) and (fj, w^ . . .) certain finite 

 and determinate functions U, V in such a way that U-f V 

 and U\' may be l>y the sorm€ ride of correspondence 

 associated with the sequences (A) and (B) above. 



Among the various ways in which this can be done,. 

 M. Borel's method of exponential summation is par- 

 ticularly interesting. Briefly it is this : let 



;/(a) = «„-f?(,a-H '-1=^-1-: 



then the function 



3! 



j= / e-"ii{a)da 



is defined to be the exponential sum of the series- 

 "o + "i + ''3 + - • • • When 5//-„ is convergent, .e coincides 

 with the sum in the ordinary sense ; the important point 

 is that J' may be finite even when 2;/„ is divergent ; the 

 series is then " e.xponentially summable " — aisolurclysfy, if 



= / e~" I ii{a) I da 



is a convergent integral. M. Borel proves that (in the 

 case of absolute summability) if U, V are the expon- 

 ential sums oi'S.u,, and ^v„, then U-l-Y and UV are the 

 exponential sums of the series whose terms are given, 

 under (A) and (B) above ; in other words, the formal 

 laws of rational operation are satisfied. In a similatr 

 sense,an absolutely summable series may be diflTerentiated 

 any number of times. 



As an example of the practical value of these results, 

 suppose we have a differential equation V'{y,y, y", . . .) = o 

 in which y,y,y" . ■ . enter rationally: then, if this is 

 found to be formally satisfied by a series S/i-,,, which, 

 although divergent in the ordinary sense, is exponentially 

 summable, the exponential sum is actually a solution of 

 the differential equation. 



In Chapter iv. M. Borel applies the idea of exponential 

 summation to an interesting problem in function-theory. 

 -Suppose we have a power-series 



«,, ;■ H,.V -I- u., .\~+ . . . 

 which is convergent within a circle of finite radius, but 

 divergent outside of it. Within the circle, this series 

 defines a function of .r, say /(.r) ; within the same region 

 the series is exponentially summable, and its sum is /{x). 

 Hut the exponential sum may exist and be finite in a 

 region /iirger ih?in the circle of convergence of the power- 

 series ; in this case the exponential sum is an analytical 

 continuation of/(.t) outside the circle, and the new region 



