LONDON. Oct. 2--The trunk of a woman was found in a recess of the new police offices on the Thames embankment this afternoon. The head, arms and legs were missing. The remains, which were in an advanced state of decomposition, were wrapped in rough cloth and tightly bound with cord. It is believed that the arms recently found at Pimlico and Lambeth were cut from this body.

LONDON, Oct 2.--The excitement occasioned by the latest Whitechapel murders increased rather than abated yesterday and to-day. The police have made no progress in their efforts to establish a clue to the murderer and have further lost public confidence in their effeciency by the fact that the persons arrested on suspicion of connection with the tragedies were released to-day for want of even a scintilla of evidence upon which to justify their retention in custody. They profess to be hopeful of success, however, and many people are inclined to hope with them in the absence of any suspicion in their own mind, even remotely defined, as to the criminal's motive, calling or identity. The private rewards offered for his arest added to that of the Lord Mayor bring the total sum up to £1200, and it is likely that it was soon reach £200.

Dr. Forbes Winslow and Sir Riedon Bennett and other medical experts express themselves as more than ever convinced that the perpetrator of the crimes is a homicidal lunatic. The fact that the latest murders were committed within the jurisdiction of the city police has created a sharp rivalry between that body and Gen. Warren's metropolitan detectives. Many wagers have been laid upon the question of the murderer's capture, with the odds largely in favor of the city police force solving the mystery and landing the fiend in jail. It is now proposed to send a circular to every householder in the East End requesting the details of the movements, habits, etc. of their respective lodgers. The theory has largely obtained support to-day that the murderer has two domiciles, one probably as a shop or surgery, to which he retires for the purpose of removing the traces of his crime, and the other a dwelling house where he is known as a lodger, whose habits and manners make him a desirable tenant in the estimation of his landlady. The police are being urged to station bloodhounds in the Whitechapel district, in the hope that they may be able by their keenness of scent to run the fiend down when he seeks a fresh victim.

An inquest was held to-day on the body of the woman found murdered in a narrow court off Berners street Monday morning. A sister of the victim was called and deposed that she awoke at 12 o'clock Monday morning and heard kisses and a sound which she thought was made by a person falling to the ground. She was convinced that her sister was dead, and after reading the accounts of the murder in the newspapers, went to the morgue and recognized the body of the murdered woman as that of her sister. The house in which the witness resides is several miles from Berners street. The murder is believed to have been committed about 12.50 [30?] o'clock Sunday morning.